ABSTRACT
Title of Dissertation: CLOSED AFFINE MANIFOLDS
WITH AN INVARIANT LINE
Charles Yves Daly
Doctor of Philosophy, 2021
Dissertation Directed by: Professor William Goldman
Department of Mathematics
Radiant manifolds are affine manifolds whose holonomy preserves a point.
Here we discuss certain properties of closed affine manifolds whose holonomy pre-
serves an affine line. Particular attention is given to the case wherein the holonomy
acts on the invariant line by translations and reflections. We show that in this case,
the developing image must avoid the translation invariant line providing a general-
ization to the well known fact that closed radiant affine manifolds cannot have their
fixed points inside the developing image. We conclude by generalizing this result to
translation and reflection invariant proper subspaces of the holonomy.
CLOSED AFFINE MANIFOLDS
WITH AN INVARIANT LINE
by
Charles Yves Daly
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2021
Advisory Committee:
Professor William Goldman, Chair/Advisor
Professor Richard Wentworth
Professor Scott Wolpert
Professor Christian Zickert
Professor David Mount
? Copyright by
Charles Yves Daly
2021
Dedication
To my mom and dad, who have given me a life full of opportunity, love, and
so much more.
ii
Acknowledgments
I would not be writing this without the many people in my life who have looked
after me. Professor Goldman, thank you for your constant patience and academic
guidance throughout my career. To the Portmans and Marlows, I remember each
holiday visit spent together enjoying each other?s company. You?ve always been
guiding examples of kindness in my life. To Andy, Brittany, Carl, Christophe, D-
Benz, Fill, Jasonia, Kareem, Manderson, Matt, Philip, and each of your families,
I cannot put to words the debt of gratitude I hold to each and every one of you.
Lauren, I am so thankful for all the years we get to spend together and I cannot
wait to see where life takes us. How lucky I am to spend it with you. O?Sharn and
Anatoly, thank you for helping raise me, inspiring me with imagination, and always
keeping me in your thoughts. What a joy it?s been to be your little brother. Finally,
to mom and dad, I am so indescribably proud and lucky to be your son. You have
sacrificed so much to give me a life of opportunities, hope, happiness, and love.
Thank you for every skateboard, music recommendation, guiding hand, life lesson,
hug when I needed one, piece of advice, home cooked meal, and the unconditional
love I have been blessed with in our family. Thank you all. I am so very lucky and
I hope you all know just how deeply and truly I love you all.
iii
Table of Contents
Dedication ii
Acknowledgements iii
Table of Contents iv
List of Figures v
Chapter 1: Introduction 1
Chapter 2: Affine Manifolds and Developing Pairs 5
2.1 Affine Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Affine Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Examples of Affine Manifolds . . . . . . . . . . . . . . . . . . . . . . 11
2.4 (G,X)-manifolds and Developing Pairs . . . . . . . . . . . . . . . . . 19
2.5 Parallelism and Geodesics on Affine Manifolds . . . . . . . . . . . . . 38
Chapter 3: Radiant Manifolds 41
3.1 Radiant Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Incompleteness of Radiant Manifolds . . . . . . . . . . . . . . . . . . 50
Chapter 4: Parallel Flow 58
4.1 Holonomy with an Invariant Vector . . . . . . . . . . . . . . . . . . . 58
4.2 Complete Parallel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Chapter 5: Affine Manifolds with an Invariant Line 75
5.1 Incomplete Affine Manifolds with an Invariant Line . . . . . . . . . . 75
5.2 Foliations of Affine Manifolds with an Invariant Line . . . . . . . . . 81
5.3 Affine Manifolds with an Invariant Line of Translation . . . . . . . . 85
Bibliography 92
iv
List of Figures
2.1 Square undergoing some affine transformations . . . . . . . . . . . . . 7
2.2 An affine manifold with coordinate charts . . . . . . . . . . . . . . . 10
2.3 A Euclidean torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Square undergoing some projective transformations . . . . . . . . . . 18
2.5 Developing map construction . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Developing map with homotopic paths . . . . . . . . . . . . . . . . . 28
3.1 Three dimensional Hopf-manifold . . . . . . . . . . . . . . . . . . . . 43
3.2 Developing map of Hopf-torus . . . . . . . . . . . . . . . . . . . . . . 44
3.3 Some geodesics on the Hopf-torus . . . . . . . . . . . . . . . . . . . . 45
3.4 Thurston and Sullivan?s projective structure on a torus . . . . . . . . 48
3.5 Developing map that is not a covering onto its image . . . . . . . . . 49
3.6 Radiant flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 Mapping torus of an irrational rotation on a disk . . . . . . . . . . . 65
4.2 Induced action on the disk from a quotient . . . . . . . . . . . . . . . 66
4.3 Parallel flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.1 A foliation of the universal cover of the Hopf-torus . . . . . . . . . . . 83
5.2 Foliation of the Hopf-torus with two closed leaves . . . . . . . . . . . 84
v
Chapter 1: Introduction
This thesis is concerned with affine structures on closed manifolds, in particu-
lar consequences of having proper invariant subspaces of the corresponding holonomy
representation of the manifold in question. Stated in relatively elementary terms,
affine manifolds are manifolds equipped with a preferential atlas of charts into affine
space, An, whose change of coordinates are the restriction of affine automorphisms
of An. Several examples of varying complexity are provided in Section 2.3.
Manifolds of this nature are of interest as one may conduct affine geometry
upon them. Familiar concepts from affine geometry such as convexity, parallelism,
and geodesics may all be defined on these manifolds through the use of a construction
called a developing pair. This construction is defined for a much larger class of
manifolds known as (G,X)-manifolds in Section 2.4. With this construction in mind,
some basic properties and definitions are reviewed such as completeness of a (G,X)-
structure and the correspondence between locally homogenous closed Riemannian
manifolds and discrete co-compact subgroups of G acting freely on X.
Chapter 3 provides some background on a subclass of affine manifolds known
as radiant manifolds. These are affine manifolds whose holonomy representation
preserves a point in affine space. Examples are provided and in particular an example
1
from Thurston and Sullivan is provided wherein the corresponding developing map
is not a covering onto its image [ST83]. These pathological examples of when the
developing map fails to be a covering map onto its image showcase some of the
complexities that may arise even in the low-dimensional cases. While Nagano and
Yagi showed that in the two dimensional case, the developing map of a closed affine
manifold is a covering onto its image, this statement is no longer true in higher
dimensions [NY74]. In John Smillie?s thesis, he constructs an affine structure on the
three-torus whose developing map fails to be a covering onto its image providing a
counter example to a conjecture of Thurston which stipulated that the developing
map of a closed affine manifold is a covering onto its image [Smi77].
The following theorem is a well-known fact regarding the incompleteness of
closed radiant affine manifolds.
Theorem 1.1. Let M be a closed radiant manifold. Then fixed points of the affine
holonomy are not contained within the developing image. In particular this means
the radiant structure on M is incomplete.
We reprove this theorem with a new argument that makes use of the complete
radiant flow associated to the vector field on Rn given by R = ?yi?/?yi. Lemma
3.5 shows that if one can lift the radiant flow associated to R through a local
diffeomorphism F : N ?? Rn, then F may be promoted to a global diffeomorphism
if 0 ? F (N). With this lemma, we show if a fixed point of the holonomy lies
inside the developing image, the developing map must be a diffeomorphism and the
holonomy acts trivially, contradicting the compactness hypothesis.
2
Chapter 4 investigates the consequences of having a fixed vector in the linear
holonomy of a closed affine manifold. The condition of having a fixed vector in the
linear holonomy provides an R-action on the manifold M with parallel flow. In fact,
this condition provides us with ?large? open subsets of the universal cover of M for
which the developing map restricted to these open subsets is a diffeomorphism onto
its image. These open subsets are invariant with respect to the induced R-action
on the universal cover of M and map to open cylinders under the developing map.
This result is stated in formality below and its proof is largely taken from a preprint
with some modifications and change in notation [Dal20].
Theorem 1.2. Let M be a closed affine manifold whose linear holonomy fixes a
vector in Rn+1. Then there exists a complete parallel flow on M which lifts to the
universal cover M? . In addition, for any point in the universal cover, we may find a
neighborhood of the point saturated with respect to this flow such that the developing
map restricted to these neighborhoods are diffeomorphisms onto their images.
This theorem is further generalized to the situation wherein the linear holon-
omy admits a k-dimensional subspace upon which it acts by translations in Theorem
4.2.
Chapter 5 is concerned with a generalization of radiant manifolds. Whereas
radiant manifolds preserve a single point, Chapter 5 investigates manifolds that pre-
serve an affine line. Whereas the affine holonomy may only act trivially to preserve
a point in the radiant case, the affine holonomy may act by translations, scalings,
reflections, and their compositions to preserve an affine line. Fried, Goldman and
3
Hirsch proved that there are no complete closed affine manifolds with reducible
holonomy [FGH81]. We prove the following mild generalization in Chapter 5. Let
G be the group of affine transformations preserving some fixed affine line.
Theorem 1.3. Let ? be a connected open subset of R ? Rn containing the line
R?{0} ? R?Rn. There does not exist a subgroup ? ? G with the discrete topology
acting on ? both properly and freely with a compact quotient.
After this theorem is established, Section 5.3 returns to the general study of
closed affine manifolds with an invariant line. Combining the results of Theorem
1.1 and Theorem 1.2, we prove the following theorem.
Theorem 1.4. Let M be an (n+ 1)-dimensional closed affine manifold with n ? 1
whose holonomy admits an invariant line. If the holonomy acts on the invariant
line by translations, then the developing image cannot meet this invariant line.
The corresponding result about when the affine holonomy admits an invariant
k-plane upon which the holonomy acts by reflections and translations is addressed in
Corollary 5.2. This result provides a partial affirmation of a conjecture of Goldman
and Fried which states for closed affine manifolds, proper invariant subspaces that
are acted upon unipotently by the holonomy cannot meet the developing image.
4
Chapter 2: Affine Manifolds and Developing Pairs
It is worth establishing for the purpose of clarity, this thesis works largely in
the category of smooth manifolds. Unless otherwise stated, one may assume that all
such topological spaces and maps are smooth. In addition, most of these manifolds
are assumed to be connected unless otherwise stated. That said, throughout this
thesis, the word space may be taken to mean smooth connected manifold and the
word map may be taken to mean smooth map.
2.1 Affine Space
In this section we recall some basic definitions of affine space. Let n be some
non-negative integer.
Definition 2.1. Affine n-space is a space equipped with a free and transitive Rn-
action. In other words, affine n-space is an Rn-torsor.
Fix an affine space An. For each point p ? An denote the action of the
vector v ? Rn on p by p + v. By prospect of the fact that Rn acts both freely and
transitively on An, for each two points p, q ? An, there is a unique vector v ? Rn
so that p+ v = q. This vector is occasionally denoted by qp. If one picks an origin
p ? An, then one may identify An with Rn via the map that takes each q to qp.
5
This identification in turn provides An with a vector space structure isomorphic
to Rn. The identification is non-canonical and depends on the choice of an origin.
While each such identification only differs by choice of an origin, it nevertheless
depends on picking a point. That said, there is no natural identity element of affine
space. This is in contrast to the case of a vector space that has a naturally defined
origin, namely the zero vector.
An affine homomorphism A is a map between affine spaces A : An ?? Am
such that for each p ? An there exists a linear map L : Rn ?? Rm where for each
v ? Rn we have A (p+ v) = A(p) + L(v). It is worth noting that this linear map is
well defined independent of choice of p.
For if q ? An is another point in An and M is another linear map satisfying
the above criterion, then for each v ? Rn we have that
M(v) = A(q + v)? A(q) = A (p+ (qp+ v))? A (p+ qp)
= L(qp+ v)? L(qp) = L(v) (2.1)
Note that this means that for all p, q ? An we have the vector taking A(q) to A(p),
A(q)A(p), satisfies A(q)A(p) = L(qp). In fact, this is yet another means of defining
an affine homomorphism.
The group of affine automorphisms is defined to be the group of invertible
affine homomorphisms on the space An. This group is denoted by Aff(n). Some
elements of Aff(2) acting on a square in A2 may be seen in Figure 2.1.
Up to isomorphism, this group is given by the semi-direct product of GL(n,R)
6
Figure 2.1: A figure of the affine group acting on the center dark blue square to
produce several affinely equivalent shapes in the plane. All these shapes differ
by elements of the affine group, namely through translation, reflection, rotation,
sheering, scaling, or some composition of them. Note how even though lengths and
areas are distorted, parallel lines are sent to parallel lines. This illustrates how
the affine group does not preserve the standard Euclidean metric, but nevertheless
preserves a notion of parallelism which is addressed in Section 2.5.
acting on Rn in the natural fashion. This may be seen by choosing an origin p ? An
and identifying An with Rn via the map taking q to qp. From this identification, we
associate to each affine automorphism A : An ?? An its linear part LA ? GL(n,R)
and its translational part T := A(p)p ? RnA , the unique vector taking p to A(p).
Associate to the affine automorphismsA,B ? Aff(n) their linear parts LA, LB ?
GL(n,R) and their translational parts T , T ? RnA B . Composition of the two affine
7
automorphisms obeys the equalities in Equation 2.2
( ) ( )
BA (p+ v) = B A(p) + L(A(v) = B )p+ (TA + LA(v) )
= B(p) + LB TA + LA(v) = B(p) + LB(TA) + LBLA(v) (2.2)
Equation 2.2(, the compositio)n of BA has linear part LBA = LBLA and translational
part TBA = B(p) +LB(TA) p = TB +LB(TA) which is the product of (LB, TB) and
(LA, TA) in the semi-direct product GL(n,R)nRn. Thus, we have the isomorphism
between Aff(n) and GL(n,R)nRn taking A to (LA, TA) ? GL(n,R). We frequently
write the group GL(n,R) nRn as Aff(n,R).
Affine geometry is the study of An and its invariants under the group of affine
automorphisms, Aff(n). This is in contrast to Euclidean geometry which is the
study of Euclidean space, En, and its invar?iants under the group of isometries that
preserve the Euclidean inner product x?y = xiyii . Whereas the group of Euclidean
isometries preserves notions such as distance, angle, and volume, these invariants
are lost in the affine group. In fact, perhaps the most important invariant of affine
geometry is that of parallelism which is defined in Section 2.5. Figure 2.1 shows the
affine group preserving the parallel sides of a square.
2.2 Affine Manifolds
In this section we provide some definitions and fundamentals regarding affine
manifolds. A smooth manifold is a topological manifold whose coordinate charts ?i :
Ui ?? Rn enjoy the property that their coordinate changes ? ? ??1i j are restrictions
8
of smooth maps from Rn ?? Rn. A recurring theme of geometry is to insist these
coordinate changes lie inside some group that preserve some structure in Rn, such
as the Euclidean inner product. In so doing, one may pullback the quantity via the
coordinate charts to endow the manifold with such a structure. This for example
is of natural interest in the case where the coordinate changes lie inside the group
Aff(n,R) as the standard connection ? on Rn is invariant under this group. Due
to this invariance, we may pullback the standard connection ? via the coordinate
charts to endow M with a connection modeled on affine space. This is perhaps
reasonable motivation for taking affine manifolds into consideration, as we know
every manifold admits a connection, but the choice of this is highly arbitrarily and
usually relies on some partition of unity [Lee91, Prop 4.5].
Definition 2.2. An affine structure on an n-dimensional manifold M is a collection
of charts ?i : Ui ?M ?? An so that for each connected component of V ? Ui ? Uj
there exists an affine automorphism A n ni,j : A ?? A so that ? ? ??1i j : ?j (V ) ??
?i (V ) is equal to Ai,j restricted to ?j (V ). In terms of diagrams, for each connected
component V ? Ui ? Uj we have Equation 2.3 below commutes.
V
?j ?i (2.3)
?j(V ) ? ?i(V )A 1i,j=?i??j
9
Figure 2.2: Here we have two overlapping coordinate charts ?i : Ui ?? A2 and
? : U :?? A2j j on M with connected components V,W ? Ui ? Uj. As depicted,
?j(V ) and ?i(V ) differ by a translation in the affine plane, whereas ?j(W ) and
?i(W ) differ by a dilation and rotation by ?-radians.
10
2.3 Examples of Affine Manifolds
In this section we provide several examples of affine structures on manifolds.
Example 2.1. Let ? be a discrete subgroup of Aff(n,R) acting both properly and
freely on Rn. Since the action is both free and proper, one may form the quotient
manifold M := Rn/? [Lee03, Thm 21.10]. As Rn is simply connected, the associated
quotient map p : Rn ??M is the universal covering for the manifold M .
This defines an affine structure on M as each point m ? M is evenly covered
by a neighborhood U so that p?1(U) is a collection of disjoint open subsets {Ui} in
Rn projecting diffeomorphically onto U . For each i, let pi be the restriction of p to
Ui onto U , pi : Ui ?? U . Define a chart for U into Rn by ?i : U ?? Ui ? Rn via
?i = p
?1
i .
Let W ? U ? V be a connected component of the intersection of two charts
?i : U ?? Ui and ?j : V ?? Vj where Ui and Vj are subsets of p?1(U) and p?1(V )
respectively that are projected diffeomorphically onto U and V respectively. Then
for each x ? ?j(W ), the coordinate changes are given by
( ) ( )
?i ? ??1j (x) = p?1i ? pj (x) = ?(x) (2.4)
where ? is some element of ? taking Uj to Ui. This shows the coordinate changes
are indeed locally affine.
A standard example of an affine structure arises by taking ? ? Aff(2,R) to be a
lattice of translations and forming the quotient R2/?. This quotient is diffeomorphic
11
to the two-torus. As the group of translations belongs to the isometries of Euclidean
space, the Euclidean metric passes down to the quotient providing the two-torus with
a flat Euclidean structure by which we mean the coordinate changes are locally the
restrictions of Euclidean isometries. Figure 2.3 below illustrates this construction.
Figure 2.3: Fundamental domains of a lattice of translations generated by the vectors
(1, 0) and (1, 2) acting on R2. The quotient of R2 by this lattice of translations
provides the two-torus with a Euclidean structure.
This should be contrasted with Example 3.1 which is an example of a similarity
structure. In fact the Hopf-manifolds defined in Example 3.2 all provide similarity
structures whose coordinate changes lie inside the similarity group.
Before continuing it is useful to have the notion of an affine map between
affine manifolds.
12
Definition 2.3. Let M and N be n-dimensional affine manifolds. Let F : M ?? N
be a local diffeomorphism. F is called an affine map if for each pair of charts
? : U ? M ?? An and ? : V ? N ?? An, we have that F is locally affine
in the sense that the restriction of ? ? F ? ??1 to each connected component of
? (U ? F?1(V )) is the restriction of an affine automorphism on An.
Example 2.2. Let M have an affine structure. Then each covering space p : C ??
M is naturally endowed with an affine structure such that the covering map is an
affine map. For each point c ? C, choose an evenly covered neighborhood U of p(c).
Shrink this neighborhood U sufficiently small so that there?s a chart ? : U ?? An.
Since c is contained in a unique connected component of V ? p?1(U) so that p|V is
a diffeomorphism onto U , we may compose p with the chart ? to obtain a chart for
C, ? := ? ? p : V ?? An.
This provides an affine atlas for C. Let ?i : Vi ?? X and ?j : Vj ?? X be
two patches in C and W ? Vi ? Vj be a connected component of their intersection.
By construction ?i = ?i ? p and ?j = ?j ? p for some patches ?i : Ui ?? X and
?j : Uj ?? X in M .
Let pi : Vi ?? Ui denote the restriction of p to Vi. The coordinate changes of
the patches ?i and ?j for any x ? ?j(W ) are given by Equation 2.5 below.
( ( ))
(? ? ??1i j )(x) = ((?i ? pi) ?)(?j ? p )
?1
j (x) = ? p p
?1 ??1i i j j (x)
= ? ? ??1i j (x) (2.5)
By hypothesis, since M is affine, ?i ? ??1j is locally the restriction of some affine
13
automorphism, and thus so too is ? ?1i ? ?j .
This provides our covering space C with an affine structure induced by M .
Now choose coordinate charts ? : V ?? X in C and ? : U ?? X in M where
? = ??p. Let W be a connected component of ? (V ? p?1(U)). For each x ? ?(W )
we have that ( )
? ? p ? ??1 (x) = ? ? p ? (? ? p)?1 (x) = x (2.6)
Thus, locally, in these charts, the coordinate expression of p is the restriction of an
affine map, namely the identity. Since in these charts the coordinate representation
is affine, the same holds true for any such charts, as coordinate changes are also
affine. Thus p : C ??M is indeed an affine map in the sense of Definition 2.3.
Example 2.3. In a similar spirit to Example 2.1, consider an n-dimensional affine
manifold M and let ? be a group of affine automorphisms acting on M both properly
and freely. The quotient space M/? inherits a natural affine structure such that the
projection map p : M ??M/? is an affine map.
As the projection is a covering space, for each point x ?M/?, we may choose
an evenly covered neighborhood x ? U with p?1(U) equal to the disjoint union of
{Ui} where p restricted to each Ui is a diffeomorphism onto U . Shrink Ui if necessary
to assume we may find a chart ?i : U ?? Ani . The composition ?i := ?i ? p?1i :
U ?? An serves as a chart about x ? U where p?1i : U ?? Ui denotes the inverse
of p onto Ui.
Given two charts ?i : Ui ?? An and ?j : Uj ?? An, let W ? Ui ? Uj be a
connected component of their intersection. By construction, for each x ? ?j(W ) we
14
have that
(
? ? ??
) ( ) ( )?1 ( ( ))1
i j (x) = ?(i ? p
?1
i ?) ? ? p
?1 (x) = ? p?1j j i i ? p ?1j ?j (x)
= ? ???1i j (x) (2.7)
where ? ? ? is some element of ? taking Uj to Ui. Equation 2.7 shows that the
change of coordinates ?i ? ??1j is locally given by the composition of ?j???1i . As
? : M ?? M is an affine map as defined by Definition 2.3, this means that the
expression ?i ?? ???1 n nj : A ?? A , in appropriately chosen charts, is the restriction
of an affine automorphism of An as desired. Hence M/? supports an affine structure.
To show p : M ?? M/? is an affine map choose charts ?i : Ui ?? An for
M and ?i : U ?? An for M/?. For each x ? ?i (Ui ? p?1(U)), the coordinate
representation of p is given by
( ) ( ) ( )
? ? p ? ??1 (x) = ? ? p?1 ?1i i i i ? ?i ? p?1i (x) = x (2.8)
which is affine. Since the coordinate expression of p is locally affine in these charts,
it is locally affine in all such charts, and thus p is an affine map in the sense of
Definition 2.3 as claimed.
Example 2.4. Let M and N be affine manifolds of dimension m and n respectively.
The product manifold M ?N supports an affine structure. This may be seen in the
following fashion. Note that the product of two affine space, Am?An is isomorphic
to the affine space Am+n. Given two affine automorphisms F of Am and G of An,
15
the product of these two maps is also an affine automorphism on Am+n. Another
way of putting this is that Aff(m,R)?Aff(n,R) is a subgroup of Aff(m+n,R). Let
(A, v) ? Aff(m,R) and (B,w) ? Aff(n,R).
The map taking this pair of affine automorphisms to the affine automorphism
of Am+n given by Equation 2.9 below
?? ?? ?
(A?B, v ? w) := ?? A 0 ?????? v ??? (2.9)
0 B w
is easily verified to be an injective group homomorphism.
Thus, given any two coordinate charts ? : U ? M ?? Am and ? : V ?
N ?? An one may form the product chart ? ? ? : U ? V ? M ? N ?? Am+n.
This defines an atlas of charts U ? V ?? Am+n on M ? N . By construction, the
coordinate changes are given by product of two affine maps which is affine as shown
by Equation 2.9.
Example 2.5. Let M be an affine manifold and A : M ?? M be an affine au-
tomorphism of the affine structure on M . Consider the affine product structure
on R ?M where R has the standard affine structure induced by an identification
of A1 and R. Let ? be the cyclic group generated by the affine automorphism of
R?M taking (t,m) ? R?M to (t+ 1, Am). This action is easily seen to be both
free and proper on R?M , and thus by means of Example 2.3, (R?M) /? has an
affine structure. Thus the mapping torus of an affine automorphism supports an
affine structure. This construction is known as the parallel suspension of an affine
automorphism A : M ?? M . More details on this construction may be found in
16
Goldman?s text [Gol21, p. 118].
Before proceeding we introduce the notion of a projective structure. Projective
manifolds are a larger class of manifolds which include all affine ones. Intuitively
this comes from the fact that affine space, An, sits inside projective space, RP n,
and every affine automorphism induces a projective automorphism. That is to say
that projective geometry is an extension of affine geometry, much like how affine
geometry is an extension of Euclidean geometry. This extension is illustrated in
Figure 2.4.
Definition 2.4. A projective structure on an n-dimensional manifold M is a col-
lection of charts ?i : Ui ? M ?? RP n so that for each connected component of
V ? Ui ? Uj there exists a projective automorphism P : RP n ?? RP ni,j so that
? ???1i j : ?j (V ) ?? ?i (V ) is equal to Pi,j restricted to ?j (V ). In terms of diagrams,
for each connected component V ? Ui?Uj we have Equation 2.10 below commutes.
V
?j ?i (2.10)
?j(V )
P =? ???
?i(V )1
i,j i j
As mentioned above if one picks an origin and identifies An with Rn, then
Rn ? RP n via the inclusion sending v ? Rn to the point [v : 1] ? RP n. Moreover,
each affine automorphism (A, v) ? Aff(n,R) induces a projective automorphism in
PGL(n+ 1,R) via the (n+ 1)? (n+ 1) matrix acting on RP n as shown in Equation
2.11.
17
Figure 2.4: A figure of the projective group acting on the central dark blue square
to produce several projectively equivalent shapes. This figure should be contrast
with Figure 2.1. Technically this is only an affine patch of RP 2, but nevertheless
conveys how the projective group contains the affine transformations, and more.
The red, tan, and light blue shapes are projectively equivalent to the dark blue
center square, but are not affinely equivalent it. Note that unlike the affine group,
the projective group fails to preserve parallelism as illustrated by the fact that the
parallel sides of the dark blue square are not sent to parallel sides of the light blue
square. Nevertheless the projective group still takes lines to lines.
?? ??? A v ??? [x : t] = [Ax+ vt : t] (2.11)
0 1
It is for this reason we say that projective geometry is an extension of affine
geometry as both affine space and its group of transformations sit inside projective
space and its group of projective transformations.
18
Example 2.6. Let R3 have the Minkowski metric given by (x, y) = x1y1 ? x2y2 ?
x3y3. Define P as the locus of points (x, x) = 1 where x > 0 in R3 with the pullback
metric induced by the inclusion map P ?? R3. This is the hyperboloid model of
the hyperbolic plane whose orientation preserving isometry group is SO+(2, 1).
The map P ?? RP 2 sending each point (x, y, z) in P to its homogenous
coordinates [x : y : z] ? RP 2 is easily seen to be an embedding. This inclusion is
equivariant with respect to the homomorphsim SO+(2, 1) ?? PSO(2, 1) and shows
that every hyperbolic two-manifold has a projective structure. This may be easily
generalized to higher dimensions, but for familiarity?s sake has been restricted to
the two dimensional case.
This extension of geometries means that one may construct hyperbolic struc-
tures on say for example a closed orientable genus two surface to obtain non-affine
projective structures. The fact that these are not affine follows by Benze?cri?s The-
orem which states that the Euler characteristic of a two-dimensional closed affine
manifold vanishes [Ben59]. Samual Ballas, Jeffrey Danciger, and Gye-Seon Lee have
done work to provide examples of projective structures that are not hyperbolic on
certain three manifolds [BDGS18].
2.4 (G,X)-manifolds and Developing Pairs
As one can see Definition 2.2 and Definition 2.4 are very similar in nature. In
fact, they fall under the much larger class of manifolds known as (G,X)-manifolds.
Here G is a Lie group that acts on a model space X. Here we assume the action
19
of G is strongly effective in the sense that if g ? G such that there exists an open
subset U ? X for which g|U = id|U , then g = 1.
With this context in mind, we define a (G,X)-structure on a manifold in the
following fashion.
Definition 2.5. Let G be a Lie group acting strongly effectively on a connected
manifold X. A (G,X)-structure on an n-dimensional manifold M is a collection of
charts ?i : Ui ?M ?? X so that for each connected component of V ? Ui?Uj there
exists a transformation g ?1i,j : X ?? X so that ?i ? ?j : ?j (V ) ?? ?i (V ) is equal
to gi,j restricted to ?j (V ). In terms of diagrams, for each connected component
V ? Ui ? Uj we have Equation 2.12 below commutes.
V
?j ?i (2.12)
?j(V ) ? ?i(V )g 1i,j=?i??j
In this language, an affine structure is a (Aff(n),An)-structure, and a Eu-
clidean structure is an (Isom(n),En)-structure, where Isom(n) is the isometry group
preserving the Euclidean metric. A similarity structure is a (Sim(n),An)-structure,
where Sim(n) is the group of similarity preserving transformations acting on affine
space. Finally a projective structure is a (PGL(n+ 1),RP n)-structure.
If we pick an origin in affine space and identify it with Rn we have the following
inclusions of the groups below.
Isom(n,R) < Sim(n,R) < Aff(n,R) < PGL(n+ 1,R) (2.13)
20
with Rn ? RP n via the inclusion as defined in the paragraph after Equation 2.10.
Due to these inclusions, we say that similarity structures are a generalization of Eu-
clidean structures and affine structures are a generalization of similarity structures.
Projective structures contain all of these and even hyperbolic structures as seen in
Example 2.6.
A result of Bieberbach classified Euclidean structures on closed manifolds
[Wol11]. As it so happens, these manifolds are entirely classified by an associated
representation of their fundamental group into the group of isometries of Euclidean
space. Remarkably there are only finitely many of them and also the manifolds are
finitely covered by Euclidean tori. On the other hand, David Fried classified simi-
larity structures on closed manifolds [Fri80]. These manifolds include the Euclidean
structures, but in addition, have non-Euclidean structures. Example 3.1 provides an
example of this sort of possibility. In fact, in his work, Fried showed that these are
essentially the only other types of similarity structures one obtains, namely finite
quotients of Hopf-manifolds.
Both of these results rely heavily on the use an indispensable tool associated
to (G,X)-structures called a developing pair. We now endeavor to describe this
construction.
Let M be a (G,X)-manifold. Pick a point p ? M and base the fundamental
group there. As M is path-connected, we frequently suppress the base point and
simply write ?1(M) for the fundamental group. Let p ? U be a coordinate patch,
? : U ?? X and ? : [0, 1] ?? M be any path beginning at p. We assign to this
path a point in X in the following fashion.
21
Cover the image of the path ? by (k + 1)-coordinates patches ?i : U ?? X
for 0 ? i ? k beginning with U0 = U and so that consecutive coordinate patches
overlap, meaning Ui ? Ui+1 6= ?.
Pick a mesh of times 0 = t0 < t1 < . . . < tk < tk+1 = 1 in [0, 1] so that
?(t?) ? Ui?1 for all ti?1 ? t? ? ti for i = 1 . . . k. Note this necessitates that
?(ti) ? Ui?1 ? Ui for each 1 ? i ? k. Define ?i = ?|[t ,t for each i = 0, . . . , k.i i+1]
While describing this process in full detail, it is worth following along to Figure
2.5 to get a feel for the construction. We proceed by inductively defining paths ?i
in X.
Define ?0 := ?0?0.
Let V1 ? U0 ?U1 be the connected component of U0 ?U1 containing ?(t1) and
g0,1 be the element of G so that g0,1| ?1?1(V1) = ?0 ? ?1 : ?1 (V1) ?? ?0 (V1). This
element exists by definition of being a (G,X)-manifold, and moreover is unique by
prospect of G acting strongly effectively on X.
Define ?1 := g0,1 (?1?1). The initial point of ?1 is the terminal point of ?0.
To see this note that by definition, ?1(t1) = g0,1 (?1?1(t1)). Additionally, ?1?1(t1) ?
?1 (V1), as ?1(t1) ? V1. Since g | = ? ? ??10,1 ?1(V1) 0 1 and ?1 (?1(t1)) ? ?1 (V1), this
means,
( )
?1(t1) = g0,1 (?1?1(t1)) = ?0 ? ??11 (?1?1(t1)) = ?0?1(t1) = ?0?0(t1)
= ?0(t1) (2.14)
Equation 2.14 shows the initial point of ?1 equals the terminal point of ?0 as claimed.
22
Let V2 be the connected component of U1?U2 containing ?(t2) and g1,2 be the
unique element of G so that g1,2| ?1?2(V2) = ?1 ? ?2 : ?2 (V2) ?? ?1 (V2).
Define ?2 := g0,1g1,2 (?2?2). The initial point of ?2 is the terminal point of
?1. To see this note that by definition ?2(t2) = g0,1g1,2 (?2?2(t2)). Additionally,
?2?2(t2) ? ?2 (V2), as ?2(t2) ? V2. Since g1,2|?2(V2) = ?1 ? ??12 and ?2?2(t2) ? ?2 (V2),
we have
( )
?2(t2) = g0,1g
?1
1,2 (?2?2(t2)) = g0,1 ?1 ? ?2 (?2?2(t2)) = g0,1 (?1?2(t2))
= g0,1 (?1?1(t2)) = ?1(t2) (2.15)
Equation 2.15 shows the initial point of ?2 equals the terminal point of ?1 as claimed.
Inductively define and construct (k + 1)-paths in X, ?0, ?1, ?2, . . . , ?k in the
fashion above with ?i = g0,1g1,2 . . . gi?1,i(?i?i) for 1 ? i ? k each with the property
that the terminal point of ?i?1 is the initial point of ?i. We may concatenate these
paths together to obtain the path
?0?1 . . . ?k = (?0?0) (g0,1 (?1?1)) (g0,1g1,2 (?2?2)) . . .
(g0,1g1,2 . . . gk?1,k (?k?k)) (2.16)
This path begins at ?0(0) = ?0?0(0) = ?(p) where ? : U ?? X is the original
coordinate patch. We assign to the path ? in M based at p the terminal point of
the path as defined in Equation 2.16. Figure 2.5 illustrates this construction with
three charts.
23
Figure 2.5: Here the path ? is covered by three coordinate charts, U0, U1, and U2.
The path ? is broken up into ?0, ?1, and ?2 in red, yellow, and green respectively. The
Ui?s are illustrated below the covering of ? along with their corresponding images
under each chart ?i. This figure shows how to construct the path in X given by
Equation 2.16. Starting with the image of ?2 in ?2(U2), one maps ?2?2 back under
the unique g1,2 taking ?2(V2) to ?1(V2) where V2 is the component containing ?1?s
end point. This yields the yellow and green curve in the second row of charts. We
pull this path back by g0,1, the unique element taking ?1(V1) to ?0(V1) where V1
is the component containing ?0?s end point. This yields the red, yellow, and green
path starting from the original chart ?0 : U0 ?? X. The end point of this path is
the developing map applied to ?.
24
At this moment we have a map from the space of smooth paths based at p in
M to the model space X. Denote this map applied to the path ? by dev(?). This
construction appears to depend on several choices including for example, the path ?
in question, and both the choice and number of coordinate patches used along with
their corresponding meshes of times chosen.
In regards to the covering of ? by coordinates patches, we show that this
construction depends only on the choice of the original coordinate patch ? : U ?? X
and not subsequent patches chosen in the covering.
To this end, let {Ui} be a covering of ? by (k + 1)-coordinate patches into
X such that Ui ? Ui+1 6= ? with U0 = U as before. Insert an additional covering
patch ? : W ?? X somewhere after the original coordinate patch ? : U ?? X so
that the set {U0, . . . , Ui,W, Ui+1, . . . , Uk} covers ?. Insert an additional s so that
0 = t0 < . . . < ti < s < ti+1 < . . . tk+1 = 1 and satisfies the property that the image
of ? on each interval is contained in the corresponding coordinate patch covering ?.
In particular this means ?(t?) ? W for all s ? t? ? ti+1.
At the (i + 1)-th step of the construction applied to our refined cover and
mesh, let Wi+1 be the connected component of Ui ?W containing ?(s) and hi+1 be
the unique element of G so that h|?(W ) = ? ?1i ? ? : ?(Wi+1) ?? ?i+1 i(Wi+1).
Similarly at the (i+ 2)-th step, let Wi+2 be the connected component of W ?
Ui+1 containing ?(ti+1) and j being the unique element of G so that j|?i+1(W =i+2)
? ? ??1i+1 : ?i+1(Wi+2) ?? ?(Wi+2).
At the (i+1)-th step of the original construction with the unrefined cover and
mesh, Vi+1 is the connected component of Ui ? Ui+1 containing ?(ti+1) and gi,i+1 is
25
the unique element of G so that g ?1i,i+1|?i+1(V ) = ?i ? ?i+1 : ?i+1(Vi+1) ?? ?i+1 i(Vi+1).
We claim Wi+1?Wi+2 6= ?. By the old construction, ?(t?) ? Ui for all ti ? t? ?
ti+1. By the new construction, ?(t
?) ? W for all s ? t? ? ti+1. Thus ?(t?) ? Ui ?W
for all s ? t? ? ti+1. Thus there is a path entirely contained in Ui ?W containing
both ?(s) and ?(ti+1), so they are in the same connected component of Ui ? W .
Since ?(s) ? Wi+1 ? Ui ?W , this means ?(ti+1) ? Wi+1, thus ?(ti+1) is in both
Wi+1 and Wi+2 showing their intersection is non-empty.
Since ?(ti+1) ? Vi+1 ? Wi+2, and ?(ti+1) ? Wi+1 ? Wi+2, we may choose a
sufficiently small connected open subset in their intersection, call it C.
The composition of hj is defined on all of X, but we may restrict it to the
domain ?i+1(C) ? ?i+1(Wi+2). Because C ? Wi+1 ?Wi+2, this means hj restricted
to ?i+1(C) is given by
( ) ( )
hj = ?i ? ??1 ? ? ? ??1i+1 = ? ? ??1i i+1 (2.17)
Similarly since gi,i+1|? = ? ? ??1 , and ? (C) ? ? (V ), this meansi+1(Vi+1) i i+1 i+1 i+1 i+1
gi,i+1|? (C) = hj? (C). By prospect of the fact that G acts strongly effectively oni+1 i+1
X, and both hj and gi,i+1 agree on the open subset C, this means hj = gi,i+1.
In the old construction, the developing map applied to the path ? with the
cover {Ui} and mesh of {ti} is given by the terminal point of Equation 2.16. Taking
the terminal point of the path in Equation 2.16 yields the expression in Equation
2.18.
dev(?) = g0,1g1,2 . . . gi?1,igi,i+1gi+1,i+2 . . . gk?1,k (?k?k(1)) (2.18)
26
On the other hand, in the new construction, the developing map applied to
the path ? with the refined cover and refined mesh provides a new path. Taking the
terminal point of this path yields the expression in Equation 2.19
dev(?) = g0,1g1,2 . . . gi?1,ihjgi+1,i+2 . . . gk?1,k (?k?k(1))
= g0,1g1,2 . . . gi?1,igi,i+1gi+1,i+2 . . . gk?1,k (?k?k(1)) (2.19)
The second equality follows by Equation 2.17. Thus the developing map applied to
this refined cover is left unchanged. Consequently, given any two covers {Ui} and
{U ?j} of ? with meshes {ti} and {sj}, we may take the union of these two covers and
meshes to obtain a refined cover and mesh of both.
Since we showed the developing map is left unchanged on a refinement by an
additional chart and additional time in the mesh, induction shows the developing
map is left unchanged on the refinement on their unions. Thus the construction
applied to {Ui}, {ti} and {U ?j}, {sj} on ? provide the same terminal point in X, so
this construction is well-defined for any choice of cover or mesh of times.
That said, one can also show that if ? and ? are two paths in M that are
homotopic relative endpoints, then dev(?) = dev(?). The technicalities of this
proof are somewhat involved, the basic idea is as follows and taken from Goldman?s
text [Gol21, p. 99]. If ? and ? are paths homotopic relative endpoints in M and
contained in the original coordinate chart ? : U ?? X, then clearly both ? and ?
develop to the same point in X.
Now let h : [0, 1]2 ?? X be a homotopy between ? and ? relative endpoints
27
so that h(t, 0) = ?(t) and h(t, 1) = ?(t). This homotopy can be broken up into
?small? homotopies, namely homotopies such that we may find meshes of times
0 = s0 < s1 < . . . < sm = 1 so that ?s and ?j s are homotopic relative endpointsj+1
so that one may find a cover and mesh of both paths {Ui} and {ti} so that both
?s (ti) and ?s (ti) are in the same connected component of Ui?1 ? Ui. Figure 2.6j j+1
provides an example of this.
Figure 2.6: Here two homotopic paths, relative endpoints, are provided where each
endpoint of each segments is contained in the same connected component. If we
denote the new path as ?, then both ?(t1), ?(t1) ? V1 and ?(t2), ?(t2) ? V2. As both
? and ? are covered by the same charts, and each ?(ti) and ?(ti) are contained in
the same connected component of Ui?1 ? Ui, this guarantees the change of coordi-
nate elements gi?1,i are the same for all i in Equation 2.18, thus yielding the same
developing image.
Given such a cover one can then show both these paths develop to the same
28
point, and by induction, ? and ? both develop to the same point, so the developing
map is well defined on the homotopy classes of paths based at p ? M . In fact,
this construction is provided in full detail for a similar process regarding the Baker-
Campbell-Hausdorff formula [Hal04, p. 76].
Consequently the developing map is well defined on the universal cover of M .
We denote this map by dev : M? ?? X. As this map is defined on the space of
homotopy classes of paths based at p ? M , it is natural to ask how the developing
map changes if one considers the concatenation of a path ? beginning at p with a
loop ? based at p.
To this end, let {Ui} and {ti} be a cover and mesh for ? and {Vj} and {sj}
be a cover and mesh for ?. One may form the concatenation ?? and combine the
covers and meshes appropriately to obtain the cover {U0 = V0, . . . , Vm, U0, . . . , Uk}
and mesh 0 = s0/2 < s1/2 < . . . < sm+1/2 = 1/2 = (t0 + 1)/2 < (t1 + 1)/2 < . . . <
(tk+1 + 1)/2 = 1. Following Equation 2.18, the developing map applied to the path
? will yield
dev(?) = h0,1h1,2 . . . hm?1,m (?m?m(1)) (2.20)
where ?j : Vj ?? X are the charts covering ?. Thus, when the developing map
is applied to the concatenation ?? with the covering and mesh as described above,
one has the result as in Equation 2.21.
dev(??) = (h0,1h1,2 . . . hm?1,m) (g0,1g1,2 . . . gk?1,k (?k?k(1))) = hdev(?) (2.21)
29
where ?i : Ui ?? X are the charts covering ?. The expression h is the fixed element
in G corresponding to the product h0,1 . . . hm?1,m which depends only on the choice
of ?.
In fact, since we showed the developing map is well defined on homotopy
classes of paths, this yields a group homomorphism hol : ?1(M) ?? G that takes
the homotopy class of ? to the element h as defined in Equation 2.21. Given how
the fundamental group acts on the universal covering space of M via deck trans-
formations, we obtain the fundamental relation between the developing map and
holonomy below in Equation 2.22. For each [?] ? ?1(M) which acts on M? by the
corresponding deck transformation, the diagram below commutes for any such [?].
[?]
M? M?
(2.22)
dev dev
X X
hol [?]
The developing map is easily seen to be a local diffeomorphism, as locally it is given
by the original coordinate patch ? : U ?? X or some translate of it by an element of
G as seen by Equation 2.18. The corresponding pair (dev, hol) is frequently referred
to as a developing pair for the (G,X)-structure on M .
Thus to each (G,X)-structure on M , we have a means of assigning a local
diffeomorphism from M? into X and a group homomorphism from the fundamental
group into the group G of transformations of X that obey the commutative diagram
in Equation 2.22 above.
On the other hand, one may start off with a simply connected space N and a
30
discrete group ? acting both properly and freely on N to form the quotient manifold
M := N/?. Given a group homomorphism hol : ? ?? G, and a hol-equivariant
local diffeomorphism dev : N ?? X, this defines a (G,X)-structure on M .
Coordinate patches ? : U ?? X may be constructed in the following fashion.
Take an evenly covered open neighborhood U ?M and the disjoint collection {Ui}
of its inverse image under the projection p : N ?? M . Define ?i : U ?? X via
dev ? p?1i where pi := p|U : Ui ?? U similar to Example 2.3. Given a connectedi
component W ? U ? V of overlapping charts ?i : U ?? X and ?j : V ?? X, note
that for any x ? ?j(W ), we have
( )?1
(?i ? ??1j )(x) = (dev ? p?1i ) ? dev ? p?1j (x)
= dev ? p?1i pj(n) = dev(?n) = hol ? dev(n) = (hol ?) (x) (2.23)
where n is the unique element in Uj ? N so that dev(n) = x and ? is the element
in ? taking Uj to Ui. Thus Equation 2.23 shows that locally the coordinate changes
are given by elements of G, as hol ? ? G, and consequently, M supports a (G,X)-
structure as claimed.
The observation above provides us with a means of constructing developing
maps via passing to the universal cover of an open subset of X equipped with a free
and proper discrete subgroup of G acting on it.
Lemma 2.1. Let ? ? G be a discrete subgroup acting on a connected open subset
? ? X both properly and freely. The quotient manifold M := ?/? inherits a (G,X)-
structure whose developing map dev : ?? ?? X has ? as its developing image and
31
? as its holonomy image. In this case, the developing map is a covering onto its
image.
Proof. As ? is an open subset of X, it is easily seen to have a natural (G,X)-
structure. One can easily generalize Example 2.3 to show the quotient manifold
M := ?/? inherits a (G,X)-structure such that the projection p : ? ?? M is an
(G,X)-map.
Let U be an evenly covered neighborhood of M and let Ui ? ? ? X be a
corresponding chart associated to the inverse image of U under p?1. If we denote
pi as the restriction of p to each chart Ui, then the collection ?i := p
?1
i : U ?? X
serve as charts for U . Here, as in Example 2.1, the charts ?i : Ui ?? X are simply
given by the identity restricted to Ui. Let W ? U ? V be a connected component
for two charts ?i : U ?? X and ?j : V ?? X. For each x ? ?j(W ), the change of
coordinates is given by
( ) ( )
?i ? ??1j (x) = p?1i ? pj (x) = ?(x) (2.24)
where ? ? ? ? G is some element of ? taking Uj to Ui.
Pick a base point p ? M and a point q ? ? above it. Let y ? ?. We claim
that y is in the developing image of M . Pick a path ? based at q to y contained in
?. Cover ? with (k + 1)-charts {Vi} ? ? so that p : ? ?? M restricted to each Vi
is a diffeomorphism onto a chart for M . Choose times {ti} so that ?(ti) ? Vi?1 ? Vi
for each 1 ? i ? k. Denote ?i = ?|[t ,t ]. The charts Ui := p(Vi) cover the pathi i+1
? := p(?) in M which is based at p ?M and ?(ti) ? Ui?1 ? Ui.
32
If one applies the developing map construction as in Equation 2.18, one can
show that the curve ? based at p develops to the curve ? based at q. This follows
because the original chart U0 containing p takes p to q via p|?1V : U0 ?? V0 ? X.0
Since each Vi?1 ? Vi 6= ?, the change of coordinates gi?1,i in Equation 2.16 or ? in
Equation 2.24 are all trivial. Hence ? develops to the curve ?. In particular, this
means dev(?) is the terminal point of ?, which is by construction y and is therefore
part of the developing image. On the other hand, every path in M based at p lifts
to a unique path based at q in ?. Thus every path in M develops to a point in ?,
so the developing image of M equals ?.
That the holonomy is ? as claimed may be seen in the following fashion.
Equation 2.24 shows that every element of the holonomy is some element of ?.
That every element of ? is realized as a holonomy element may be seen as follows.
A standard fact of algebraic topology yields that ? is isomorphic to ?1(M)/p??1(?)
where p : ? ??M is the covering map projection [Hat01, Prop 1.40]. The isomor-
phism is defined by sending homotopy classes of loops [?] ? ?1(M, p) to the unique
deck transformation taking q ? ? to the terminal point of the unique lift of ? to ?
based at q. That said, let [?] ? ?1(M, p) be a class of loops based at p whose image
under the isomorphism is given by ? ? ?.
Now let ? be a path based at p ?M and dev(?) be its developing image which
we know by the above arguments is the terminal point of the unique lift of ? to ?
based at q ? ?. Denote this lift by ?? so dev(?) = ??(1).
Consider the concatenation ??. Again, we know that dev(??) is the terminal
point of the unique lift of ?? to ? based at q ? ?. In particular, this means the
33
latter half of the lift of ?? is a lift of ? based at the terminal point of the lift of ?
based at q, which is by construction, ?q. By the uniqueness of paths, this means
the latter half of the lift of ?? is the unique lift of ? based at ?q. Thus the latter
half of the lift of ?? at q is equal to ???. In particular, their endpoints agree, so
dev(??) = ?dev(?). Hence ? is an element of the holonomy as claimed.
That the developing map is a covering onto its image follows as the developing
map was shown to take paths in M based at p to the terminal point of its lift to
? based at q. Passing to the universal cover, we get dev : M? ?? X that takes
homotopy classes of paths in M based at p to the terminal point of their lifts to ?
based at q which, up to identification of M? and ??, is precisely the standard projection
from the universal cover ?? ?? ?. Hence the developing map is a covering onto its
image.
Returning to the general theory of (G,X)-structures, we introduce the most
elementary examples of (G,X)-manifolds, namely, complete (G,X)-manifolds.
Definition 2.6. Let M be a (G,X)-manifold with X simply connected. We say
that M is complete if and only if the developing map is a covering onto X.
In the case where X is simply connected and the developing map is a covering
onto its image, one may recover the manifold from the holonomy alone. In fact,
since dev : M? ?? X is a covering onto X, by the uniqueness of universal covers,
the developing map may be promoted to a global diffeomorphism. Hence M?/?1(M)
is diffeomorphic to X/hol?1(M). Thus in the complete case, the holonomy alone
defines the structure.
34
It is worth noting that in general, the holonomy fails to determine the (G,X)-
structure on a manifold. For example, Goldman constructed complex projective
structures on the torus with the same holonomy representation but different devel-
oping maps [Gol87].
That said, there are certain conditions that guarantee a (G,X)-structure is
complete. For example if M is a closed manifold and X is a simply connected
Riemannian manifold with a G-invariant metric gX , then every (G,X)-structure on
M is complete.
One may pullback gX on X via the developing map to a Riemannian metric
gM? on M? so that the developing map dev : M? ?? X is a local isometry. The holon-
omy invariance of gX guarantees that pullback metric is invariant under the deck
transformations, and thus descends to a Riemannian metric on M . By compactness,
the Hopf-Rinow Theorem guarantees this metric is complete, and thus so too is the
metric gM? [Lee91, Thm 6.13].
From here one may apply the the fact that a local isometry on a complete Rie-
mannian manifold into a connected Riemannian manifold is a covering map [KN63].
Thus the developing map is a covering from M? to X, and by uniqueness of the
universal cover, is necessarily a diffeomorphism. Thus the corresponding (G,X)-
structure on M is complete.
In this context, the study of complete (G,X)-structures may be reduced to
the study of discrete subgroups ? of G acting freely on X whose quotient X/? is
compact.
The existence of a developing pair provides some interesting topological con-
35
straints on certain (G,X)-manifolds. For example, one can show that if X is non-
compact, then the fundamental group of a closed (G,X)-manifold is necessarily
infinite. This follows from the simple observation that if M is compact with a finite
fundamental group, then its universal cover M? is also compact. So if M did indeed
support a (G,X)-structure, one may choose a developing pair, under which dev M?
is compact, and therefore closed in X. Additionally, since the developing map is
a local diffeomorphism, it is in particular an open map, and thus dev M? is also an
open subset of X. By connectedness of X, this means dev M? = X, contradicting the
fact that M? is compact. In particular, this prohibits say for example, the projective
plane from admitting an affine structure.
As was shown in Example 2.2, to each affine structure on a manifold M ,
there is a natural affine structure on every covering space of M . This example is
easily generalized to each covering space of a (G,X)-structure. That said, if we let
p : C ?? M denote the projection map, and pick points y ? C above p ? M , and
base their fundamental groups there, we have two developing maps devC : C? ?? X
and devM : M? ?? X. Up to the identification of C? and M? , these two maps are
equal.
Lemma 2.2. Let M be a (G,X)-manifold and p : C ?? M be a covering map.
Let C inherit the (G,X)-structure induced by M . Identifying C? with M? , the corre-
sponding developing maps are equal.
Proof. Let ? denote a path based at y ? C. We may choose coordinate charts
{Vi} induced from the (G,X)-structure on M to cover ? in such a fashion that p
36
restricted to each such chart is itself a chart in the original atlas for M . This may
be done by construction as seen in Example 2.2. That is to say, the charts covering
? are of the form ?i := ?i ? p : Vi ?? X where ?i : Ui ?? X are charts from the
(G,X)-structure on M .
From this we obtain the developing map applied to ?. By Equation 2.18 it
will be of the form
devC(?) = g0,1g1,2 . . . gk?1,k (?i?i(1)) (2.25)
where each gi?1,i is the induced by the coordinate changes of ? ? ??1i?1 i .
We may project the curve ? to the curve ? := p ? ? in M based at p. This
curve is covered by the charts ?i : Ui ?? X by construction. The developing map
applied to the curve ? will yield
devM(?) = h0,1h1,2 . . . hk?1,k (?i?i(1)) (2.26)
Equation 2.5 provides us that the coordinate changes of ?i and ?i are the same,
and thus hi?1,i = gi?1,i for all i. Moreover, ?i?i(1) = (?i ? p)?i(1) = ?i?i(1). Using
the identification of the universal cover of C with M? via homotopies of paths in C
based at y and homotopies of paths in M based at p, we see the induced developing
map of M and C on M? are identical. Consequently devM(?) = devC(?).
37
2.5 Parallelism and Geodesics on Affine Manifolds
Equipped with the notion of a developing pair we may define familiar notions
of affine geometry on affine manifolds. Let I denote an open interval in R. Recall
that a curve ? : I ?? An is a geodesic if and only if ???(t) = 0 for all t ? I.
Identifying An with Rn, this definition is equivalent to saying that ?(t) = x+ tu for
some x, u ? Rn.
The notion of being a geodesic is invariant under the group of affine transfor-
mations. Note if (A, v) ? Aff(n,R), as in the notation from Equation 2.2, then
d2 d2
(A, v)(x+ tu) = A(x+ tu) + v = 0 (2.27)
dt2 dt2
This fact allows us to define the notion of a geodesic on an affine manifold.
Definition 2.7. Let M be an affine manifold and pick a developing pair dev : M? ??
An and hol : ?1(M) ?? Aff(n,R). A curve ? : I ??M is a called a geodesic if and
only if its lift to M? develops to a geodesic in An under the developing map.
At first glance it may appear that Definition 2.7 depends on the choice of lift
of ? to the universal cover. But if we choose a lift ?? of ?, any other lift will differ
by an element of the fundamental group. That is to say that another lift ? will be
equal to [?]?? where [?] ? ?1(M) acts on M? by deck transformations. Thus we have
dev (?) = dev ([?]??) = hol [?]dev (??) (2.28)
38
Since hol [?] ? Aff(n,R), this means by Equation 2.27 that dev (?) is also a geodesic.
Consequently Definition 2.7 is well defined independent from the choice of lift of ?
to M? .
Similarly one may define a notion of parallelism in An. Two subsets X, Y ? An
are parallel if and only if they differ by a translation. That is to say there exists a
vector u ? Rn so that X+u = Y . The notion of parallelism is yet another invariant
of the affine group.
Take an affine automorphism A : An ?? An with associated linear part L :
Rn ?? Rn. For each y ? Y , we have that A(y) = A(x + u) = A(x) + L(u) by
definition of an affine map. Thus AY = AX +L(u), and consequently AY and AX
are parallel as well.
This fact allows us to define parallelism of curves on an affine manifold.
Definition 2.8. Two curves ? and ? on an affine manifold M are said to be parallel
if and only if their lifts develop to parallel curves in affine space.
Again, this notion is well defined for different lifts will differ by elements of
the fundamental group acting by deck transformations. By hol-equivariance of the
developing map and the fact that affine transformations take parallel sets to parallel
sets as seen in the paragraph above, this provides us with a well defined notion of
parallel curves in M .
In fact, we may combine these two notions to obtain the notion of parallel
flow which will be referenced again later in Chapter 4. Recall a complete flow on
a manifold M is a collection of diffeomorphisms ?t : M ?? M such that for all
39
t, s ? R, we have that ?t+s = ?t ??s where ?0 = idM . In other words, a complete flow
is group homomorphism from R into the diffeomorphism group of the manifold M .
For the purpose of this thesis, all flows are assumed to be complete unless otherwise
stated.
Definition 2.9. A flow on an affine manifold M is said to be a parallel flow if and
only if its flow lines are parallel and geodesics.
We provide a very simple example of this on the Euclidean torus.
Example 2.7. Consider R2 and the group of translations ? generated by (1, 0) and
(0, 1). The quotient is a Euclidean torus. The R-action given by translation along
a non-zero direction of u ? R2 taking the points (x, y) to (x, y) + tu descends to the
torus. This defines a parallel flow on the Euclidean torus.
40
Chapter 3: Radiant Manifolds
3.1 Radiant Manifolds
In the exploration of affine manifolds, a very natural sort of affine manifold
to take into consideration is a radiant manifold. These are manifolds whose affine
holonomy preserves a point. Manifolds of this nature have been explored in thorough
detail by several authors such as Suhyoung Choi and Thierry Barbot [Cho01] [BC01].
We begin by providing a definition.
Definition 3.1. An n-dimensional affine manifold is radiant if and only if its affine
holonomy fixes a point in affine space. Equivalently, its affine holonomy is conjugate
to a subgroup of GL(n,R).
The most elementary of example of a closed radiant manifold is the Hopf-circle.
This is a non-Riemannian affine structure on S1. Some details of this example are
provided below in Example 3.1.
Example 3.1. Let X = R+ and let ? be the cyclic group generated by the dilation
acting on X taking each x to ?x for some positive ? 6= 1. It is clear that ? acts both
freely and properly on X, and its quotient is a circle. By Lemma 2.1, this provides
S1 with a radiant structure which we refer to as the Hopf-circle.
41
This structure is in contrast to the standard Euclidean structure on S1 which
is the quotient of the real line acted upon by a cyclic group of translations. The fact
they are inequivalent may been seen as the Hopf-circle is geodesically incomplete.
In fact, every geodesic on the circle becomes undefined in a finite amount of time.
For example let ?(t) : (0, 1) ?? X be the geodesic ?(t) = 1 ? t. It descends
to a geodesic ?(t) : (0, 1) ?? X/? under the quotient map p : X ?? X/?. This
geodesic, by construction, cannot be extended beyond any t ? 1 for if so then its
lift to X beginning at x = 1, which is ?, would be definable beyond the origin.
This sort of behavior showcases the difference between Euclidean and affine
structures on closed manifolds. By the Hopf-Rinow Theorem, closed Euclidean
manifolds are all geodesically complete. On the other hand the Hopf-circle provides
an example of an affine structure on a compact manifold which is not geodesically
complete.
Example 3.2. Let n ? 2 and X = Rn \ 0 and ? be the cyclic group generated
by dilation acting on X taking each x to ?x, for some positive ? 6= 1 much like in
Example 3.1. Again ? acts both properly and freely on X, though here the quotient
is a trivial S1-bundle over Sn?1. Figure 3.1 below illustrates this for the case of
n = 3.
Manifolds that arise from this construction are known as Hopf-manifolds. The
case where n = 2 is known as a Hopf-torus and is of particular interest in both
Nagano and Yagi?s and Oliver Baues? classifications of the flat affine structures on
the two-torus [NY74] [Bau14]. In Figure 3.2 below, one can see the fundamental
42
Figure 3.1: Here we illustrate some fundamental domains of the action of ? on X.
Three concentric spheres that all differ by a scaling by ? are drawn. A fundamental
domain may be taken to be either region bounded by the spheres along with the
bounding spheres themselves. Points on an outer sphere are identified with points
on the inner sphere by the scaling factor ?, thus the quotient is seen to be the trivial
S1 bundle over the base S2. The surrounding box is purely cosmetic.
43
Figure 3.2: Here are some fundamental domains of R2 \ 0 which is acted upon by a
cyclic group of dilations. Each annulus forms a fundamental domain for the Hopf-
torus where the outer circle gets identified with the inner circle by dilation. By
Lemma 2.1, R2 \ 0 is also the developing image of this corresponding structure on
the torus.
domains corresponding to the action of dilation on R2 \ 0.
Unlike Example 3.1, the Hopf-torus has many complete geodesics. Simply take
any geodesic in R2 \ 0 that does not point towards the origin. It will descend in the
quotient to a geodesic which exists for all such time. That said, for each point on the
manifold, there exists a direction for which a geodesic in this particular direction
cannot be extended beyond a finite amount of time as seen in Figure 3.3. This
can be seen by picking any lift of a point on the torus to x ? R2 \ 0. The unique
direction connecting the origin and x is the deficient direction for which geodesics
44
pointed towards the origin cannot be defined for all such time. This direction is
well-defined as other lifts are of the form ?nx.
Figure 3.3: As the green geodesic is defined for all time, it will descend to a complete
geodesic on the Hopf-torus. This is in contrast with the red geodesic pointed towards
the origin that will in a finite amount of time become undefined, and thus the Hopf-
torus is geodesically incomplete.
Example 3.3. Let M be a projective manifold of dimension n. Pick a developing
pair dev : M? ?? RP n and a holonomy map hol : ?1(M) ?? PGL(n+1,R). We may
lift dev to a map into Sn, the double cover of RP n. We may also lift the holonomy
map to the group of lifts of PGL(n+1,R) which is GL(n+1,R)/R+ where R+ is the
subgroup of positive diagonal matrices in GL(n + 1,R). We abusively denote both
these maps as dev and hol. As GL(n+ 1,R)/R+ is isomorphic to SL?(n+ 1,R), the
45
group of matrices of determinant ?1, this yields the commutative square
[?]
M? M?
(3.1)
dev dev
Sn Sn
hol[?]
where here the holonomy map takes ?1(M) ?? SL?(n+1,R). This is the definition
of a projective structure used in Serge Dupont?s classification of projective structures
whose holonomy factors through a subgroup of the projective group isomorphic to
Aff+(1,R) [Dup00]. With Equation 3.1, we may form the trivial R+ bundle on M?
and Sn. The trivial R+ bundle on Sn is diffeomorphic to Rn+1 \ 0.
This induces a local diffeomorphism F : R+ ? M? ?? Rn+1 \ 0. Pick some
positive non-trivial scaling factor ? and define the homomorphism ??? ?? GL(n+
1,R) that takes ? ? R+ to the matrix in GL(n+ 1,R) with ??s along the diagonal.
The group generated by ??? and ? (M) acting on R+1 ? M? by their respective
actions is both proper and free. The quotient is S1?M , the product of a Hopf-circle
and the original projective manifold M . The local diffeomorphism F is equivariant
with respect to the induced holonomy homomorphism from ? (S11 ?M) ?? GL(n+
1,R) and thus provides S1 ?M with a radiant affine structure by the remarks in
the third paragraph after Equation 2.22. This example is a very specific case of the
radiant suspension of a projective manifold which is defined in greater generality in
Goldman?s text [Gol21, p. 128].
Example 3.4. This particular example is found in Thurston and Sullivan?s work
wherein they describe many interesting affine, inversive, and projective structures
46
on manifolds [ST83]. Construct a projective structure on the torus in the following
fashion. Pick three real numbers a < b < c so that a+ b+ c = 0. Form the diagonal
matrix A consisting of a, b, and c and consider the linear flow defined by etA. This
complete flow on R3 \ 0 descends to a complete flow Ft on RP 2 with three fixed
points corresponding to the coordinate axes of R3 and three invariant projective
lines defined by the coordinate planes in R3. The three fixed points of the flow in
RP 2 are a sink, source, and saddle.
Pick an affine patch of the projective plane containing the three fixed points
of the flow. Let ? : S1 ?? RP 2 be an immersed curve contained in this affine patch
bounding the source and sink that is everywhere transverse to the flow lines of Ft.
Define dev : R ? S1 ?? RP 2 by the flow applied to the immersed curve, namely
map each (t, ?) to Ft (?(?)). This map a local diffeomorphism as the immersed curve
is everywhere transverse to the flow lines. We may lift the developing map to the
universal cover of R?S1 to get a local diffeomorphism which we also abusively write
as dev : R2 ?? RP 2.
This map obeys an equivariance property. In particular, dev(t + 1, ?) =
eAdev(t, ?) where eA is acting on RP 2 by its induced projective transformation. On
the other hand, dev(t, ? + 1) = dev(t, ?) as this corresponds to the curve Ft(?(?))
wrapping along ? around back to its original position. Figure 3.4 shows illustrates
these properties.
Thus we have a homomorphism hol : ? (S11 ? S1) ?? PGL(3,R) for which
dev is hol-equivariant. By the remarks in the third paragraph after Equation 2.22,
this pair defines a projective structure on the torus.
47
Figure 3.4: The three arrowed projective lines represent the images of the coordinate
planes under the projection of R3 \ 0 ?? RP 2. They intersect at the source, sink,
and saddle of the flow induced by etA, namely the images of the coordinate axes
under the projection. This figure depicts a blue immersed curve everywhere tangent
to the flow along with a time one map of the curve under the corresponding flow.
The developing map is not a covering onto its image. In the category of
connected smooth manifolds, a local diffeomorphism is a covering map if and only
if the unique path lifting property is satisfied [Kap04]. The curve illustrated in
Figure 3.5 does not have a lift to R? S1. This follows because the projective lines
defined by the coordinate planes in R3 are invariant under the flow of Ft on the
projective plane. By starting our path on a point of ? which does not intersect the
projective lines, we know its evolution under the flow must also remain outside these
48
projective lines. Because the time component, t ? R, will diverge to infinity as it
approaches the invariant projective line, this curve does not lift through the map
dev : R? S1 ?? RP 2, nor dev : R2 ?? RP 2.
Figure 3.5: The projective arrowed lines are invariant submanifolds of the induced
flow on RP 2. Thus the red dots representing the intersection of ? with the projective
lines remain on the projective lines under the evolution of the flow. The green curve
begins at a point not on an invariant projective line. The lift of the green curve
diverges to infinity in R ? S1 as it is lifted. This follows because the evolution of
the segment of ? containing the green point under the flow remains in an invariant
region for all time. Thus the green curve does not lift to R? S1.
With this projective structure on the torus, we may take the trivial radiant
suspension of the torus as in Example 3.3 to yield an affine structure on the three-
torus. By construction this affine structure is radiant, and moreover its developing
49
map is not a covering space onto its image which is R3 minus the three coordinate
axes.
3.2 Incompleteness of Radiant Manifolds
By definition, the holonomy of radiant manifolds leaves a point invariant. It
is a well known fact that if the radiant manifold is closed, then the invariant point
in not inside the developing image. In particular this means that no closed radiant
manifold is complete. Here we present a new proof of this fact using the radiant
flow induced by the holonomy.
Theorem 3.1. Fixed points of the holonomy of a closed radiant manifold are not
contained within the developing image
The proof of this fact will be broken into several pieces. First we establish a
lemma regarding the ability to pullback vector fields by local diffeomorphisms.
Lemma 3.1. Let F : M ?? N be a local diffeomorphism. Let Y be a vector field
on N . There exists a unique pullback vector field X on M so that F?X = Y .
Proof. For each point m ? M choose an open neighborhood m ? U so(that F)
restricted to U is a diffeomorphism onto F (U). Define Xm := d (F
?1)F (m) YF (m) .
Smoothness of X follows immediately as locally X is the composition of smooth
functions, X|U = d(F | )?1U ?Y ?F |U . Both uniqueness and the fact F?X = Y follow
readily by construction.
The next lemma in conjunction with Lemma 3.1 allow us to pullback vector
50
fields invariant under the holonomy through the developing map to ?1(M)-invariant
vector fields on M? .
Lemma 3.2. Let G and H be Lie groups acting on manifolds M and N . Let
? : G ?? H be a homomorphism accompanied by a ?-equivariant immersion F :
M ?? N in the sense that the following diagram commutes for all g ? G.
g
M M
F F (3.2)
N N
?(g)
If Y is a ?(G)-invariant vector field on N and X a lift of Y in the sense that
F?X = Y , then X is also G-invariant.
Proof. For any g ? G and m ?M , we have the following equalities
dFgm (dgm(Xm)) = d(F ? g)m (Xm) = d(?(g) ? F )m (Xm) ( )
= d (?(g))F (m) (dFm(Xm)) = d (?(g))F (m) YF (m)
= Y?(g)F (m) = YF (gm) = dFgm (Xgm) (3.3)
The fourth equality follows as F?X = Y and the fifth because Y is invariant under
the action of ?(G). The first and last equality together with the fact F is an
immersion imply that dgm(Xm) = Xgm, thus X is invariant under G.
The next lemma shows that completeness of a vector field is inherited through
covering spaces. Within the proof we address the fact that if F : M ?? N is a
map for which F?X = Y for some vector fields X on M and Y on N , then F takes
51
flow lines of X to flow lines of Y . This fact will be used repeatedly throughout the
remainder of this thesis.
Lemma 3.3. Let F : C ?? N be a covering space and X and Y be vector fields
such that F?X = Y . X is complete if and only if Y is complete.
Proof. Let X be complete, and n ? N . Let ?n be the integral curve through n
corresponding to Y . Pick any point c ? C above n. Because X is complete, the
integral curve through c corresponding to X is defined for all time. Denote it by ?c.
Note that
( )
(F ? ? )?(t) = dF (??c ?c(t) c(t)) = dF?c(t) X?c(t) = Y(F??c)(t) (3.4)
Since (F ? ?c)(0) = F (c) = n, this means F ? ?c is an integral curve through n
corresponding to Y and by uniqueness of integral curves, is in fact equal to ?n.
Since n was arbitrary, Y is indeed complete.
Conversely, if Y is complete, pick c ? C. Let ?c be the integral curve through
c corresponding to X. Let ?n be the integral curve through n := F (c). By com-
pleteness of Y , ?n is defined for all time. As C is a covering space of N , there exists
a unique lift of ?n to ??c in C based at c. Similarly to Equation 3.4 we have
( ? )
Y?n(t) = ?
?
n(t) = (F ? ?? )?c (t) = dF?? (t) ??c (t) (3.5)c
( )
As F?X = Y , we have dF?? (t) X?? = Yc c(t) F??? (t) = Y?n(t). Since F is a lo-c
cal diffeomorphism, Equation 3.5 together with the previous sentence imply that
52
?
??c (t) = X?? (t) so ??c is an integral curve of X through ??c(0) = c defined for all time.c
Uniqueness of integral curves implies that ??c = ?c, and thus X is complete.
In the proof of Lemma 3.3, the situation where X is complete yields complete-
ness of Y even if F is not a covering space. This means so long as the condition
that F?X = Y is satisfied, F need only be smooth for completeness of X to imply
completeness of Y .
Next we establish a topological lemma regarding the ability to promote a local
diffeomorphism to a global one.
Lemma 3.4. Let F : X ?? Y be a local diffeomorphism where X is connected and
let U be an open subset of X such that F restricted to U is a diffeomorphism onto
Y . Then U = X, so F is a diffeomorphism.
Proof. By hypothesis U is open. If un is some sequence of elements converging to
some x ? X, we claim x ? U .
Define yn := F (un). By continuity, yn converges to F (x). Since F restricted
to U is a diffeomorphism, there exists a unique u ? U so that F (u) = F (x). We
claim un converges to u.
Let u ? V be any open neighborhood in X about u ? U . Because U is open,
we may shrink V if necessary to assume u ? V ? U . Because yn converges to
F (x) = F (u) and F (u) ? F (V ) is open in Y , as F is an open map, there exists
a sufficiently large N ? N so that F (un) ? F (V ) for all n ? N . Since F is a
diffeomorphism on U , this means un converges to both u and x. By uniqueness of
limits, this means u = x, so U both closed and open in X, thus all of X.
53
We now prove a dynamical lemma regarding the radiant flow on Rn associated
to the GL(n,R)-invariant vector field R = ?yi?/?yi.
Lemma 3.5. Let F : N ?? Rn be a local diffeomorphism where N is connected.
Assume that the radiant flow associated to the vector field R = ?yi?/?yi can be
lifted to N . By this we mean, there exists a complete R-action on N denoted R?t so
that the diagram below commutes for all t ? R.
R?
N t N
F F (3.6)
Rn Rn
Rt
If 0 ? F (N), then F is a diffeomorphism.
Proof. Let F : N ?? Rn be such a local diffeomorphism and R?t be the lifted flow
on N . Note F?1{0} is a discrete subset of N . Choose countably many disjoint open
subsets {Ui} about each point in F?1{0}. As F is a local diffeomorphism, we may
choose each Ui in such a fashion that F restricted to each Ui is a diffeomorphism
onto an open ball Bi about 0.
? Let R??Ui denote the saturation of Ui with respect to flow on N , namely
t?R R?tUi. This saturation is an open set and thus an open embedded submanifold
of N . We claim that F restricted to R??Ui is a diffeomorphism onto Rn.
By construction F is equivariant with respect to the R-actions on N and Rn.
Moreover F (U?i) is some open subset about 0 ? Rn, so its saturation with respect
to the R-action induced by R = ?yi?/?yi is all of Rn. In fact, for each y ? Rn,
we know that Rt(y) = e
?ty, so the expansion of an open ball about 0 will indeed
54
contain all of Rn. Consequently, it suffices to show that F restricted to R??Ui is
injective.
To do so, first we show that for all t > 0 we have R?tUi ? Ui. Let ui ? Ui
and consider the flow line R?tui for t > 0. Let F |U denote the restriction of Fi
to Ui. Because F (R?tui) = RtF (ui) ? Bi for all t > 0 and F |U : Ui ?? Bi is ai
diffeomorphism, this means that R?tui = F |?1U (RtF (ui)) for all t > 0. Thus R?tui ? Ui i
for all t > 0 as claimed. Figure 3.6 illustrates the radiant flow restricted to some
open subsets of N .
Figure 3.6: Here several neighborhoods U1, U2, and U3 are all mapped to open balls
containing the origin of their corresponding color. The flow lines of R are depicted
below in light blue and their corresponding lifts are depicted in light blue above.
The flow R?t for t > 0 takes Ui to a subset of itself as depicted.
With this established, let u, v ? Ui and choose s, t ? R so that F (R?su) =
55
F (R?tv). Assume without loss of generality that s ? t ? 0. By equivariance of F ,
this means F (R?s?tu) = F (v). Because s? t ? 0, the above shows that both R?s?tu
and v are in Ui and mapped to the same point in Bi. Because F was assumed
to be a diffeomorphism on Ui to Bi, this means R?s?tu = v. Now if s ? t > 0,
Rs?tF (u) = F (v) for s ? t > 0. The only point satisfying this condition in Rn is
0, as all other points have non-periodic flow lines, thus both F (u) = F (v) = 0, so
u = v because there?s only one point in Ui mapped to 0 and it is fixed under the
lifted flow so R?s?tu = v. If s? t = 0, then R?s?tu = v implies u = v. In either case
we see F is injective. Thus F is a diffeomorphism from R? n?Ui to R as claimed.
Applying Lemma 3.4 to F : R??U
n
i ?? R yields that F is a diffeomorphism
from N onto Rn as claimed.
With Lemma 3.5 established, we may easily prove Theorem 3.1.
Proof. Let M be a closed radiant manifold. By definition its affine holonomy is
conjugate to a subgroup of the general linear group. Identify An with Rn by picking
an origin and pick a developing pair dev : M? ?? Rn and hol : ?1(M) ?? GL(n,R).
The radiant vector field R = ?yi?/?yi is invariant under the holonomy. By
Lemma 3.1 and Lemma 3.2, there exists a vector field R? on M? that is invariant
under the action of ?1(M) on M? by deck transformations so that dev?R? = R.
This vector field descends to a complete vector field on M where completeness
is consequence of the fact that M is compact. Thus there is an R-action on M
which lifts to an R-action on M? by Lemma 3.3. Since dev?R? = R, this means that
the developing map is equivariant with respect to the R-actions corresponding to R?
56
and R respectively.
Now if 0 ? dev M? , we may apply Lemma 3.5 to yield the developing map is
a diffeomorphism from M? ?? Rn, thus the radiant structure on M is complete.
Thus M is diffeomorphic to Rn/hol?1(M). Because ? acts freely on M? , so too does
the holonomy group as dev is a diffeomorphism. But each element of the holonomy
preserves 0 ? Rn, and thus by freeness, the holonomy is trivial so M is diffeomorphic
to Rn contradicting the fact that M is compact.
It is worth noting that in the proof of Theorem 3.1, we constructed the lift
of the radiant vector field R = ?yi?/?yi to the universal cover which was ?1(M)-
invariant and descends to a vector field on the closed manifold M . Since the only
zero of R is at the origin, which we have shown is not an element of the developing
image, this means there are no zeros of the vector field on the closed manifold. As
this vector field is everywhere non-zero, the Euler characteristic of M vanishes. This
is related to the long standing Chern Conjecture which stipulates that the Euler class
of a closed affine manifold vanishes. Bertram Kostant and Dennis Sullivan prove
this in their paper for the case of complete affine manifolds [KS75].
57
Chapter 4: Parallel Flow
4.1 Holonomy with an Invariant Vector
In this chapter we explore the consequences of having an invariant parallel
vector field on a closed affine manifold. Let M be a closed affine (n+1)-dimensional
manifold whose linear holonomy fixes a vector v ? Rn+1. Choose a developing
pair dev : M? ?? An+1 and hol : ?1(M) ?? Aff(n + 1,R). Picking an origin in
affine space An+1 yields a coordinate system onto Rn+1. Applying the necessary
conjugation we may assume the affine holonomy sits inside the group
????? ?? ? ????
? ?
? 1 u ?????? t ?hol ? ? ?? ???
?? ?? ?t ? R, uT , v ? Rn, and A ? GL(n,R)??? (4.1)0 A v
where u is some row vector of length n, v is some column vector of length n, t is
some real number, and A is some n ? n invertible matrix with real entries. In the
identification between An+1 and Rn+1, we denote the first coordinate by x and the
remaining n-coordinates by yi where 1 ? i ? n. This labeling will be used again for
the remainder of the thesis. Note the parallel vector field ?/?x is invariant under
the holonomy group in Equation 4.1.
58
Given that the developing map is a local diffeomorphism, Lemma 3.1 yields
the existence of a vector field X? on M? so that dev?X? = ?/?x whereas Lemma 3.2
applied to the commutative square
[?]
M? M?
(4.2)
dev dev
Rn+1 Rn+1
hol[?]
guarantees that X? is ?1(M)-invariant as ?/?x is holonomy invariant.
Because X? is a ?1(M)-invariant non-zero vector field on M? , X? descends to a
non-zero vector field X on M so in particular closed manifolds of this form have
Euler characteristic zero. Much like in the argument provided for the proof of
Theorem 3.1, this vector field is complete because M is compact. In addition, the
corresponding flow on M is parallel in the sense of Definition 2.9. This may be seen
because dev?X? = ?/?x. Because flow lines of X develop to flow lines of ?/?x under
the developing map, M supports a complete parallel flow.
4.2 Complete Parallel Flow
In this section we explore the consequences of having a complete parallel flow
induced by X? on the universal cover obtained from the pullback under the developing
map of the vector field ?/?x.
Since dev?X? = ?/?x, flow lines of X? are sent to flow lines of ?/?x as seen by
Equation 3.4. In particular, this means that the developing map is equivariant with
respect to the corresponding R-actions on M? and Rn+1 induced by X? and ?/?x
59
respectively.
Note that Rn+1 may be realized as a principal R-bundle induced by the R-
action of translation along the x-coordinate. We claim that this structure may be
pulled back via the developing map so that M? may also be realized as a principal
R-bundle. Before proceeding, we establish a lemma that guarantees that we may
pullback principal bundle structures.
Lemma 4.1. Let G act on manifolds M and N and F : M ?? N be a G-equivariant
map. If G acts properly on N , then G acts properly on M . Additionally, if G acts
freely on N , then G acts freely on M .
Proof. Let G act properly on N . By the characterization of proper group actions,
it suffices to show that if mi and gi are sequences in M and G respectively where
both mi and migi converge, then so does a subsequence of gi [Lee03, Prop 21.5] .
Let mi converge to m and gimi converge to m
?. Then ni := F (mi) converges
to F (m) by continuity whereas by equivariance of F , we have that F (gimi) =
giF (mi) = gin
?
i converges to F (m ). As the action of G on N is proper, this means
that a subsequence of gi converges to g in G, so the action of G on M is proper.
To prove that freeness on N implies freeness on M , note that if gm = m for
some g ? G and m ?M , then by equivariance of F , F (m) = F (gm) = gF (m), and
if g stabilizes F (m), g must be the identity because G acts freely on N . This proves
freeness of G on M .
Note that as a consequence of Lemma 4.1, if G acts both properly and freely
on N , then G acts both properly and freely on M . Thus, the principal G-bundle
60
structure inherited from the proper and free action ofG onN pullbacks to a principal
G-bundle structure on M .
Returning to the situation at hand, the R-action on Rn+1 is translation along
the x-coordinate of the vectors in Rn+1. This R-action is both free and proper on
Rn+1 and as the developing map is equivariant with respect to both R-actions, the
R-action on M? is also free and proper by Lemma 4.1. By the preceding remarks,
this means that M? inherits a principal R-bundle structure.
As R is contractible, this means the principal bundle structure of M? is trivial
[Cal13]. In particular it admits a global cross section N ? M? . In terms of diagrams
this means that there exists a R-equivariant diffeomorphism ? : M? ?? R ? N
where the diagram in Equation 4.3 commutes. The map q1 : M? ?? M?/R is
the associated quotient map to the proper and free action R-action on M? whereas
q2 : R ? N ?? M?/R is the trivial bundle up to an identification of M?/R with N
which exists by prospect of the fact that the q1 admits a global cross section.
M? ? R?N
(4.3)
q1 q2
M?/R
Define dev? : R ? N ?? Rn+1 via dev? = dev ? ??1 and let ?1(M) act on
R?N by [?](t, n) := (? ? [?] ? ??1) (t, n). By construction, the map dev? obeys the
commutative square for all [?] ? ?1(M) as in Equation 4.4.
61
R? [?]N R?N
dev? dev? (4.4)
R? Rn R? Rn
hol[?]
We distinguish the first factor of Rn+1 = R ? Rn as it is the direction of
the parallel flow as defined by ?/?x. This new developing pair defines the same
geometric structure on M as initially endowed, but this new developing map enjoys
the property that it respects the parallel flow on M? ' R?N inherited from ?/?x.
This is made precise in Lemma 4.2.
Lemma 4.2. For any (t, n) ? R?N and s ? R, dev?(t+ s, n) = dev?(t, n) + (s, 0).
Proof. Let (t, n) ? R ? N and s ? R. Define m? := ??1(t, n). Since ?, and
consequently ??1, is R-equivariant we have
??1(t+ s, n) = ??1 (s ? (t, n)) = s ? ??1(t, n) = s ? m? (4.5)
Because the original developing map, dev, maps flow lines of X? to flow lines of ?/?x,
dev?
( )
(t+ s, n) = dev ? ??1 (t+ s, n) = dev (s ? m?) = dev(m?) + (s, 0)
= dev?(t, n) + (s, 0) (4.6)
Returning to the original action of ?1(M) on M? , recall that X? is invariant
under the action of ?1(M), so [?]?X? = X?. Consequently, if ?x? is a flow line of X?
62
beginning at x?, then [?]?x? is a flow line of X? beginning at [?]x?. In terms of the
corresponding R-action on M? , this says the action of ?1(M) on M? and the R-action
on M? commute. Symbolically, for all x? ? M? , we have that [?]t ? x? = t ? [?]x?.
Combining these observations, we have the action of ?1(M) on R?N as defined
in the paragraph after Equation 4.3 also commutes with the R-action on R ? N .
This is a simple consequence of the fact that
( ) ( )
[?]s ? (t, n) = ? ? [?] ? ??1 (s ? (t, n)) = s ? ? ? [?] ? ??1 (t, n) = s ? [?](t, n) (4.7)
where the second equality follows as the action of s commutes with ?,??1, and the
action of ?1(M) on M? .
As a consequence of Equation 4.7, this means that the action of ?1(M) acts
on the trivial fiber bundle structure of p2 : R?N ?? N . More specifically, for each
(t, n) ? R ? N , and [?] ? ?1(M), [?] (R? {n}) = R ? {n?} for some n? ? N . This
induces an action of ?1(M) on the base space N . Thus, for each [?] ? ?1(M), the
diagram in Equation 4.8 commutes.
R? [?]N R?N
p2 p2 (4.8)
N N
[?]
A similar statement holds about the holonomy acting on the trivial fiber bundle
structure of p2 : R?Rn ?? Rn. Referring to Equation 4.1, each holonomy element
is of the form
63
? ?? ?
??? 1 u ?? thol [?] = ???? ??? (4.9)
0 A v
and thus takes lines of the form R?{y} for each y ? Rn to lines of the form R?{y?}
for some y? ? Rn. Thus the holonomy action on R ? Rn descends to an action on
Rn such that the following diagram commutes
R? Rn hol[?] R? Rn
p2 p2 (4.10)
Rn Rn
hol[?]
In fact, the induced action of a holonomy element as induced by Equation 4.10 on
Rn is easily seen to be
hol [?]y = Ay + v = (A, v)y (4.11)
for any y ? Rn as seen by referring to Equation 4.9.
It is worth noting, while the action of ?1(M) on R?N is both proper and free,
this is not necessarily the case for the induced action of ?1(M) on N . The example
below illustrates the fragility of the free and proper conditions.
Example 4.1. Let ? = R ? D where D ? R2 is the open unit disk about the
origin. Let ? be the cyclic group generated by the symmetry of ? taking (t, p) to
(t+1, R?p) where R? is rotation by an irrational multiple of ?. The quotient of ?/?
is a solid open torus as seen in Figure 4.1.
Note that ? takes lines of the form R?{p} to lines of the form R?{Rn? p} for
some n ? Z so there is an induced action of ? on D.
64
Figure 4.1: A fundamental domain for the action of ? on ?. The top and bottom
disks of the solid open cylinder are identified via an irrational rotation. In particular,
the red and black points are identified respectively. The blue line segments are
included to illustrate the irrational rotation and are also identified in the quotient
?/?.
This induced action of ? on D is neither free nor proper. The group stabilizes
0 ? D, and worse yet, the orbit of every non-zero point is a dense subset of the
circle of its corresponding radius. Figure 4.2 illustrates this.
Recall that dev? : R?N ?? R?Rn enjoys the property that it is equivariant
with respect to the R-actions on R?N and R? Rn as in Equation 4.6. Thus dev?
descends to a smooth map dev? : N ?? Rn that is defined by Equation 4.12 below.
65
Figure 4.2: Here the induced action of ? on D is illustrated. The red point cor-
responds to the line of symmetry of the original cylinder and is fixed under the
induced action of ? so the action is not free. Moreover, the orbit of a black point
with radius r > 0 inside D under the induced action provides a dense subset of the
circle of radius r. Hence the induced action of ? on D is also not proper.
?
R?N dev R? Rn
p2 p2 (4.12)
N Rn
dev?
As dev? is the composition of a local diffeomorphism and smooth submer-
sion, it too is a smooth submersion from an n-dimensional manifold to another
n-dimensional manifold, so in particular it too is a local diffeomorphism. By con-
struction, dev? is equivariant with respect to the actions of ?1(M) on N and the
holonomy of Rn. Specifically, for each [?] ? ?1(M), the diagram in Equation 4.13
66
commutes.
[?]
N N
? ? (4.13)dev dev
Rn Rn
hol[?]
While Equation 4.13 looks as though it defines a geometric structure on
N/?1(M), it does not necessarily. As mentioned in Example 4.1, ?1(M) does not
necessarily act properly and freely on N .
Nevertheless, dev? is still a local diffeomorphism. In fact, to some extent we
may relate the developing map dev? and dev?. This comes as a consequence of the
fact that N embeds as a leaf in the foliation of R?N .
Lemma 4.3. Let U ? N be an open subset for which dev? restricted to U is a diffeo-
morphism onto its image. Then dev? restricted to R?U ? R?N is a diffeomorphism
onto its image.
Proof. Let U ? N be such an open subset. As dev? is a local diffeomorphism, it is
in particular an open map, and the image of dev?(R?U) is an open submanifold of
R ? Rn. Thus to show dev? is a diffeomorphism on R ? U , it suffices to show that
dev? restricted to R? U is injective.
This follows as if (t, n) and (s,m) are points in R ? U so that dev?(t, n) =
dev?(s,m), then dev?(n) = dev?(m). Because n,m ? U , and dev? is a diffeomorphism
on U , n = m. Hence (t, n) and (s, n) lie on the same flow line R? {n}.
By R-equivariance, dev? maps flow lines to flow lines. Because dev?(t, n) =
dev?(s, n), this necessitates that t = s, for otherwise this would necessitate the flow
67
line R? {n} would get mapped to a circle which is not a flow line of ?/?x.
Consequently dev? restricted to R?U is injective, and thus a diffeomorphism
onto the open subset dev?(R? U) = R? dev?(U).
Note that Lemma 4.3 implies that so long as dev? is a diffeomorphism on some
open subset U ? N , this open subset may be saturated by the parallel flow on
R ? N to yield a diffeomorphism of the developing map dev? on R ? U . We call
neighborhoods of the form R ? U parallel tubular neighborhoods. Figure 4.3 below
illustrates the saturation of the open subset U upon which dev? is a diffeomorphism
onto its image.
Figure 4.3: On the left is an open neighborhood U ? N for which dev? is a diffeo-
morphism onto its image. On the right is the saturation of both these neighborhoods
with respect to the R-actions. The map dev? takes the flow lines of R?U illustrated
in light blue, to the flow lines of R?Rn also illustrated in light blue. The slice 0?U
in red is mapped to some slice of R? dev?(U) whose slice is also illustrated in red.
With this established, we provide the proof of one of our main theorems.
68
Theorem 4.1. Let M be a closed affine manifold whose linear holonomy fixes a
vector in Rn+1. Then there exists a complete parallel flow on M which lifts to the
universal cover M? . In addition, for any point in the universal cover, we may find
a neighborhood of the point saturated with respect to the parallel flow such that the
developing map restricted to this neighborhood is a diffeomorphism onto its image.
Proof. If the linear holonomy fixes a vector in Rn+1, then up to conjugation, the
holonomy lies inside the subgroup defined by Equation 4.1. The vector field X?
that projects to ?/?x under the developing map as constructed by Lemma 3.1 and
Lemma 3.2 projects to the manifold M as it is ?1(M)-invariant. Its corresponding
complete flow is both parallel as the flow lines develop to flow lines of ?/?x which
are parallel.
Associate to the universal cover of M the fiber bundle structure R?N ?? N
along with the local diffeomorphism dev? satisfying Equation 4.13. For any point
(t, n) ? R?N , pick a neighborhood U ? N so that dev? is a diffeomorphism onto its
image in Rn. Such a neighborhood exists as dev? is a local diffeomorphism. Saturate
this neighborhood with the parallel flow to obtain the neighborhood R ? U about
(t, n). Lemma 4.3 shows the developing map is a diffeomorphism when restricted to
this neighborhood thus completing the proof.
We now begin exploring a natural generalization of the result in Theorem 4.1.
Let M be a closed affine (n + k)-dimensional manifold where k ? 1 and whose
linear holonomy fixes linearly independent vectors v1, v2, . . . , v ? Rn+kk . Similar
to the beginning of this section, choose a developing pair dev : M? ?? Rn+k and
69
hol : ?1(M) ?? Aff(n+ k,R). Applying the necessary conjugation we may assume
each element of the affine holonomy is of the form
?? ?? ??? Ik B ?????? Thol [?] = ??? (4.14)
0 A v
where Ik is the k?k identity matrix, B is some k?n matrix, 0 denotes the n?k zero
matrix, T and v are column vectors of length k and n respectively, and A denotes
an n?n invertible matrix. In the identification between An+k and Rn+k, denote the
first k-coordinates by xi and the latter n-coordinates by yj where 1 ? i ? k and
1 ? j ? n. The parallel vector fields ?/?xi are all invariant under the holonomy
group as defined in Equation 4.14.
Similar to the previous section, as each of these vector fields is invariant under
the holonomy, we obtain k-complete vector fields X?i on M? via Lemma 3.1, Lemma
3.2, and Lemma 3.3. Each flow corresponding to X?i is complete by compactness of
M and moreover, these flows commute by Lemma 4.4.
Lemma 4.4. The flows of each X?i for 1 ? i ? k defined on M? all commute.
Proof. As each X?i is complete, it suffices to show that their brackets commute. Let
x? ? M? . By definition X?i is defined such that dev?X?i = ?/?xi. By naturality of the
Lie bracket,
70
([ ] ) [ ]
d (dev)x? X?i, X?j = [dev?X?i, d]ev?X?jx? dev(x?)
? ?
= , = 0 (4.15)
?xi ?xj dev(x?)
Because the developing map is a local diffeormophism, its differential is injective,
thus Equation 4.15 implies that [X?i, X?j]x? = 0. Since x? was arbitrary, this means the
bracket of X?i and X?j vanishes and thus their flows commute.
Because their flows commute, this means the flows of the vector fields X?i
define an Rk-action on M? corresponding to the Rk action on Rn+k induced by the
vector fields ?/?xi. The developing map by construction is equivariant with respect
to these actions. As the Rk-action on Rn+k is both free and proper, Lemma 4.1
guarantees the action of Rk on M? is both free and proper thus providing M? with
a principal Rk-structure. As Rk is contractible, this means that the principal Rk-
structure on M? is trivial and in particular admits a cross-section N such that M? is
isomorphic to the trivial principal Rk-bundle Rk ?N ?? M?/Rk.
Via the principal bundle isomorphism between M? with a trivial Rk-bundle
Rk ?N ?? M?/Rk we construct the developing map dev? : Rk ?N ?? Rk ?Rn as
in the paragraph after Equation 4.3 which takes k-planes of the form Rk ? {n} to
k-planes of the form Rk ?{y} for n ? N and y ? Rn. Just as was done in Equation
71
4.4, one may form the commutative square
Rk ? [?]N Rk ?N
dev? dev? (4.16)
Rk ? Rn Rk ? Rn
hol[?]
where ? k1(M) acts on R ?N via conjugation by the principal bundle isomorphism
between M? and Rk ?N .
As the action of ?1(M) on Rk ?N commutes with each individual flow corre-
sponding to X?i, it commutes with the entire Rk-action. Thus ?1(M) acts on N just
as in Equation 4.8.
Referring to Equation 4.14, each holonomy element is of the form
?? ?? ??? Ik B ?hol [?] = ????? T ??? (4.17)
0 A v
and thus takes k-planes of the form Rk ? {y} to k-planes of the form Rk ? {y?} for
some y? ? Rn. Thus both the ?1(M) action on Rk ? N and holonomy action on
Rk ? Rn both descend to actions on N and Rn.
Arguing in an analogous fashion to the proof of Theorem 4.1, one may quotient
out the Rk-actions in Equation 4.16 as done in Equation 4.13 to obtain a local
diffeomorphism dev? : N ?? Rn so the diagram in Equation 4.18 commutes for all
[?] ? ?1(M).
[?]
N N
dev? dev? (4.18)
Rn Rn
hol[?]
72
where ?1(M) and the holonomy are acting as established in the paragraphs before
and after Equation 4.17. Analogous to Lemma 4.3 we have the following statement
regarding the injectivity of dev?.
Lemma 4.5. Let U ? N be an open subset for which dev? restricted to U is a
diffeomorphism onto its image. Then dev? restricted to Rk ? U ? Rk ? N is a
diffeomorphism onto its image.
Proof. Let U ? N be such an open subset. Just as before in Lemma 4.3, it suffices
to show that dev? is a diffeomorphism restricted to Rk ? U .
This follows as if (T, n) and (S,m) are points in Rk ? U so that dev?(T, n) =
dev?(S,m), then dev?(n) = dev?(m). Because n,m ? U , and dev? is a diffeomorphism
on U , n = m. Hence (T, n) and (S, n) lie on the same k-plane, Rk ? {n}.
By Rk-equivariance, dev? maps k-planes to k-planes. Because dev?(T, n) =
dev?(S, n), this necessitates that T = S, for otherwise this would mean the line in
Rk connecting T and S would get mapped to a circle contradicting the fact that
dev? maps k-planes to k-planes.
We conclude this section with a theorem summarizing the generalization of
Theorem 4.1.
Theorem 4.2. Let M be a closed affine manifold whose linear holonomy fixes k-
vectors in Rn+k. Then there exists k-complete parallel flows on M which lift to the
universal cover M? . In addition, for any point in the universal cover, we may find
a neighborhood of the point saturated with respect to these parallel flows such that
73
the developing map restricted to these neighborhoods are diffeomorphisms onto their
images.
Proof. Construct the k-complete commuting flows on M? associated to the lifts X?i
of ?/?xi through the developing map. Construct the principal bundle isomorphism
of M? and Rk?N and the corresponding developing pair dev? : Rk?N ?? Rk?Rn
in Equation 4.16. Quotient out by the Rk-action to get the local diffeomorphsim
dev? : N ?? Rk as in Equation 4.18. Pick a neighborhood U for which dev? is a
diffeomorphism restricted to U ? N , and saturate this neighborhood with respect
to the Rk-action. Lemma 4.5 guarantees dev? on Rk ? U is a diffeomorphism onto
its image.
74
Chapter 5: Affine Manifolds with an Invariant Line
5.1 Incomplete Affine Manifolds with an Invariant Line
We begin this section by recalling a result of Fried, Goldman, and Hirsch
wherein they proved the non-existence of certain affine space forms. In particular,
they showed that there does not exist complete closed affine manifolds with reducible
holonomy [FGH81]. We state their theorem below.
Theorem 5.1. Let M be a compact complete affine manifold. Then the affine
holonomy representation is irreducible.
Due to their result, it immediately follows that there does not exist a closed
complete affine manifold with an invariant line as such a manifold would have a
reducible holonomy.
A natural follow up question to the result is whether or not there are incom-
plete closed affine manifolds with an invariant line whose holonomy representation
is reducible. Certainly the Hopf-manifolds as defined in Example 3.2 provide such
examples with reducible holonomy, yet they fail to contain their entire invariant line.
As it so happens, this is in general an impossibility which is stated as a theorem
below, but first we establish some notation.
75
LetG be the subgroup of affine transformation of An+1 that preserve some fixed
affine line. Pick an origin on this line to identify An+1 with Rn+1. Put coordinates
on Rn+1 by (x, y) with x ? R and y ? Rn.
We may rotate Rn+1 about this origin to assume that the invariant line is
defined by the equation y = 0. Thus each affine transformation in consideration
must preserve the line y = 0. Thus, up to conjugation, we may assume that G is
equal to the subgroup of Aff(n+ 1,R) as defined below.
????????
?? ? ?
G = ??
? ?
? r v ?????? t ?????? ?? ?r =6 0, t ? R, vT ? Rn, and A ? GL(n,R)??? (5.1)0 A 0
With this established, we state our theorem.
Theorem 5.2. Let ? be a connected open subset of R?Rn containing the line y = 0.
There does not exist a subgroup ? ? G with the discrete topology acting on ? both
properly and freely with a compact quotient.
Proof. As a preliminary observation, note that no element ? ? ? may have r 6= 1.
For if so, then the action of ? on the invariant line y = 0 has a fixed point. This
follows as ?? ?? ? ? ? ? ??? r v ?????? t ??? ??.? x ??? ??? rx+ t= ??? (5.2)
0 A 0 0 0
and because rx+ t = x has a solution in x for r 6= 1, this would mean ? would not
be acting freely on R ? Rn. Hence the group ? must act by translations along the
76
invariant line so we may assume ? is a subgroup of the affine transformations of the
form
??????? ?
?
??
? ?
? ? ?
?
?? 1 vP = ?? ??
t ?????? ?? ?t ? R, vT ? Rn, and A ? GL(n,R)?
0 A 0 ?? (5.3)
The map from P ?? R defined by sending each element in Equation 5.3 to its
translational part t is a group homomorphism. Since its image is a subgroup of the
real numbers under addition, it is either a dense subgroup of R or cyclic. Because
the group action of ? is proper on R?Rn, and thus the line y = 0 in particular, the
image of the group homomorphism must be cyclic.
Because ? acts freely on R?Rn, the group homomorphism taking each element
of Equation 5.3 to its translational part t is injective. For if ? and ?? are two elements
with the same translational part then ????1 acts trivially on the invariant line and
thus by freeness ? = ??. Hence ? is generated by a single element of P which we
abusively denote by ?? ?? ?
P = ?? 1 v ?????? t ??? (5.4)
0 A 0
where v is some n-length row vector, A is an invertible n? n-matrix, and t is some
non-zero real number. The fact that t is non-zero follows because otherwise P would
stabilize the point (0, 0) ? ? contradicting the hypothesis ? acts freely on ?.
We may linearly conjugate P by an element Q of the form in Equation 5.5
77
below for any wT ? Rn. ?
??? 1 w ?
?
Q = ?? (5.5)
0 In
This yields that QPQ?1 is given by
??
?
? 1 w + v(A? In)? ?
?? ?
????? tQPQ 1 = ??? (5.6)
0 A 0
If 1 is not an eigenvalue of A, then we may find a solution in w to w+v(A? In) = 0
so that QPQ?1 leaves invariant both the n-plane x = 0 and line y = 0.
If 1 is an eigenvalue of A, let u ? Rn be an associated eigenvector. Let V denote
the subspace of R ? Rn generated by the linear combinations of (1, 0) and (0, u).
By construction V is invariant under the action of ?. The affine transformation P
applied to a(1, 0) + b(0, u) for some a, b ? R is provided in Equation 5.7.
?? ?? ?? ? ? ? ? ? ? ??? a ?? ?? a+ v(bu) ??? ??? t ??? ??? a+ b(vu)P = + = ???+??? t ??? (5.7)
bu A(bu) 0 bu 0
Equation 5.7 shows V is indeed an affine subspace of R? Rn invariant under
the action of ?. As V is a closed ?-invariant plane in R ? Rn, ? ? V a closed
embedded ?-invariant surface of ?, though not necessarily connected. Because the
action of ? is both proper and free on ? it is both proper and free on ??V . We may
form the quotient manifold (??V )/?. This quotient is compact because (??V )/?
sits inside ?/? as a closed subset of a compact space, and thus is itself compact.
78
Since ? is open in R ? Rn, ? ? V is open in V . Let C be the component of
? ? V that contains the line y = 0. Because ? preserves the invariant line y = 0,
this means ? must preserve C.
Returning to the action of P on V , identify V with R2 via the change of
coordinates ? : R2 ?? V defined by ?(a, b) = (a, bu). As seen in Equation 5.7, the
induced action of P on R2 is given by
?? ? ? ?? ? ? ?
P ?? a ??? ??? 1 vu ?= ????? a ??? ? t ?+?? ?? (5.8)
b 0 1 b 0
Under the change of coordinates, C will get mapped to an open subset of R2 con-
taining the line b = 0. Thus up to a change of coordinates, we may assume that C
is an open subset of the plane R2 containing the line b = 0, and P acts by the affine
transformation as in Equation 5.8.
As the holonomy of this compact affine manifold C/? is both abelian and
volume preserving, this means the developing map is surjective [FGH81]. By Lemma
2.1, the developing image of this structure is C ? R2 which by surjectivity is all of
R2. This though is a contradiction as if vu 6= 0, then ? does not act freely on R2.
In particular it fixes the entire line defined by b = ?t/vu.
On the other hand if vu = 0, then R2/? is an open cylinder, and thus non-
compact. Hence 1 is not an eigenvalue of A as claimed.
By the remark after Equation 5.6, we may assume that up to conjugation, P
79
is of the form ?? ???? 1 0 ?????? t ?
?
P = ?? (5.9)
0 A 0
and acts on the ?-invariant subset ? ? R? Rn.
Pick any (0, v) ? ? ? {0}?Rn with v 6= 0 which we may do as ? is open and
contains the point (0, 0). Construct the ray s(0, v) for s ? 0. We may choose an
increasing sequence of times si so the sequence si(0, v) does not converge in ?. For
if s(0, v) leaves ?, let s? be the smallest s > 0 for which s(0, v) is not in ?. This
time is achieved as ? is open. We may then simply pick any increasing sequence of
si that converges to s? to obtain the sequence si(0, v) ? ? which does not converge
nor have a limit point in ?.
On the other hand if ? contains the entire ray s(0, v) for s ? 0, simply pick
any increasing sequence si diverging to ? to obtain a sequence si(0, v) in ? that
does not have a limit point in ?.
This provides us with an infinite closed discrete subset {si(0, v)} ? ? where
no two elements are related via ?, as they all lie on the same slice of ?? ({0} ? Rn)
and thus cannot be related by ? as P has non-trivial translational part in the first
coordinate of R ? Rn. Saturate this set by the action of ? to obtain yet another
infinite discrete subset ?{(0, siv)} ? ?. We claim this set is also closed in ?. For if
there is indeed a sequence of points xj := P
nj(0, sjv) in ? that converges to a point
80
in ?, then we claim nj is eventually constant.
?
??? 0 ?
?
?? ?
? ?? ? ? ? ? ?
P nj = ?? 1 0 ?????? njt ??? .??? 0 ??? = ??? njt ??? (5.10)
s v 0 Anj 0 s v s Anjj j j v
As both components of P nj(0, sjv) must converge, in particular the sequence of real
numbers njt must converge. Because t is non-zero, njt forms a discrete subset of R.
For njt to accumulate, this means nj is eventually constant.
Since P nj(0, sjv) converges to a point in ?, and nj is eventually constant,
this means that (0, sjv) converges to a point in P
?nj? = ?. This contradicts the
construction of (0, sjv), as this set has no accumulation points in ?. Thus ?{(0, siv)}
is a closed discrete ?-invariant subset of ?.
Because ?{(0, siv)} is closed and discrete in ?, it will descend to a closed
discrete subset of ?/?. By construction, since each orbit of (0, siv) is distinct, this
means the closed discrete subset is infinite. This though contradicts the fact that
?/? is compact. Hence no such manifold can exist as originally claimed.
5.2 Foliations of Affine Manifolds with an Invariant Line
In this section we explore a natural foliation induced by an affine manifold
whose holonomy preserves an invariant line. Much like as was done in Section 5.1,
pick a point on the invariant line and identity An+1 with R?Rn. Pick a developing
pair dev : M? ?? R ? Rn whose holonomy lies inside the group G as defined by
Equation 5.1.
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The group G preserves the foliation of R ? Rn by lines of the form R ? {y}
where y ? Rn. We may pull back this foliation via the developing map to the fibers
dev?1 {y} ? M? . The connected components these fibers foliate the universal cover
of M [Lee03, p. 513].
In addition, since the holonomy group G preserves the foliation on R? Rn in
the sense that G takes leaves to leaves, the fundamental group ?1(M) also preserves
the foliation on M? . For if L is a leaf of the foliation of M? , then dev(L) ? R? {y}
for some y ? Rn. Because the developing map is hol-equivariant, this means that
dev ([?]L) ? hol [?] (R? {y}) = R? {y?} for some y? ? Rn. Thus the action of the
fundamental group preserves the induced foliation on M? and descends to a foliation
on M . This foliation obeys the property that all the leaves are parallel in the sense
of Definition 2.8. One could take this as a definition of a one-dimensional parallel
foliation in the context of affine manifolds, namely a one-dimensional foliation of
a manifold M where the leaves of the foliation develop to parallel lines under the
developing map. We conclude this section with an example of this induced foliation
on the Hopf-torus.
Example 5.1. Let M be the Hopf-torus as described in Example 3.2. Identifying
R2 with C, consider the developing map dev : C ?? C given by dev(z) = ez. The
developing map obeys the following equivariance properties, dev(z + 1) = e1dev(z)
and dev(z + 2?i) = dev(z). Thus the holonomy map takes the translations induced
by 1 and 2?i on C are sent to the diagonal 2? 2 matrix with e?s along its diagonal
and the identity map respectively.
82
Note the holonomy preserves the foliation of C by lines parallel to the imagi-
nary axes. That is to say that for any u ? R, hol [?] (R? {iu}) = R?{iu?} for some
u? ? R. In fact, if we parametrize each line by t, we may easily solve the equation
ez = t + iu. Writing z = x + iy yields that x = (1/2) ln(t2 + u2) and tan(y) = u/t
for all t 6= 0. The corresponding foliation in the universal cover, C, is given below
in Figure 5.1
Figure 5.1: One can see this foliation is invariant with respect to the translations
in the coordinate axes. Whereas most leaves of the foliations are curved, there are
leaves that are lines parallel to the x-axis. These are the leaves corresponding the
leaf R? {0} in affine space which splits into two leaves R+ ? {0} and R? ? {0} in
C? which is the developing image.
What is interesting about this foliation is that unlike the trivial foliation of
the torus by circles or the foliation of the torus by parallel translates of a line of
83
irrational slope, this foliation admits both closed and non-closed leaves. The leaves
parallel to the x-axis in the universal cover, C, descend to closed leaves in M under
the projection map p : C ?? C/?1(M) as the translations defining M are along the
coordinate axes. On the other hand, each one of the curved leaves in C projects to
a leaf of the foliation on the torus that gets arbitrarily close to the closed leaves.
Figure 5.2 depicts this behavior.
Figure 5.2: Here is the quotient of the foliation as in Figure 5.1 on the Hopf-torus.
The green leaves of the foliation correspond to lines that are parallel to the x-axis
in C whereas the blue curves correspond the curved leaves. Note the curved leaves
wrap infinitely many times around the direction defined by the green leaf, but never
meet the closed leaves.
84
5.3 Affine Manifolds with an Invariant Line of Translation
In this section we explore the consequences of having a closed affine manifold
with an invariant line whose holonomy acts on the invariant line by translations.
Similar to how closed radiant manifolds cannot have developing images containing
fixed points of the holonomy, we show that closed affine manifolds with a translation
invariant line cannot meet this line.
After showing this result we extend it to the case where the holonomy ad-
mits an invariant affine k-plane upon which the holonomy acts by translations and
reflections. This result provides partial affirmation to the conjecture of Fried and
Goldman which stipulates that proper invariant affine subspaces upon which the
holonomy acts unipotently lie outside the developing image.
Theorem 5.3. Let M be an (n+ 1)-dimensional closed affine manifold with n ? 1
whose holonomy admits an invariant line. If the holonomy acts on the invariant
line by translations, then the developing image cannot meet this invariant line.
Before proceeding, we need to first establish a technical lemma in place of
Lemma 3.1.
Lemma 5.1. Let G and H be Lie groups acting on manifolds M and N where
G is discrete and acts both properly and freely on M . Let ? : G ?? H be a
homomorphism accompanied by a ?-equivariant submersion F : M ?? N . For each
?(G)-invariant vector field Y on N , we may find a G-invariant vector field on M
so that F?X = Y .
85
Proof. For each point m ?M , because G is discrete and acts properly and freely on
M , we may find an open subset m ? U such that gU ? U = ? unless g = 1. As F
is a submersion, we may shrink U sufficiently small so that in appropriately chosen
coordinate patches, F has a coordinate representation as a linear projection, taking
coordinates (x1, x2, . . . , xn) to (x1, x2, . . . , xm) with n ? m. Thus we may construct
a smooth local lift XU of Y to U so that F?XU = Y on U .
This lift may be pushed forward on each gU to define smooth local lifts XgU :=
g?XU of Y . Each XgU is still a local?lift of Y by equivariance of F . Let V be the
disjoint union of all the gU , V := g?G gU , and define XV to be XgU on each
gU . This is well defined because gU ? U = ? for all g 6= 1. Thus XV is a local
G-equivariant lift of Y on V .
Take the union of all such V? to cover M . There exists a G-invariant partition
of unity subordinate to the covering [Wan17]. By this we mean a collection of non-
negative smooth functions {??} : M ?? R indexed by the open cover {V?} so
that
1. supp ?? ? V?
2. For each p ?M , there exists a neighborhood that intersects only finitely many
supports supp ??.
?
3. ? ?? = 1
4. Each ?? is G-invariant.
Using a ?1(M)-invariant partition of unity subordinate to the cover {V?}, we may
86
?
define the vector field X := ? ??X? which is by construction G-invariant and
smooth. Finally, we have that
(? ?? )
dFm (Xm) = dFm ??(m)X ???
? ? ( ? m) ?
= ??(m)dF ?m X? = ??(m)Ym F (m) = YF (m) (5.11)
? ?
Thus X is a G-invariant vector field of M so that F?X = Y as claimed.
We now return to the proof of Theorem 5.3.
Proof. To this end, let M be an (n + 1)-dimensional closed affine manifold whose
holonomy admits an invariant line upon which the holonomy acts by translation.
Up to conjugacy, this means the holonomy group sits inside the group G defined by
matrices of the form as in Equation 5.12
?????? ???? ???? ????1 v t ?
G = ????? ???? ???? t ? R, vT ? Rn, and A ? GL(n,R)??? (5.12)0 A 0
Pick a developing pair dev : M? ?? R ? Rn and hol : ?1(M) ?? G where G is
defined as above in Equation 5.12.
By the result of Theorem 4.1 there exists a complete parallel flow on M? defined
by the lift of the invariant vector field ?/?x through the developing map. Thus
we may assume the developing pair obeys the commutative diagram as found in
Equation 4.4. That is, we may write M? as a trivial principal R-bundle, R?N , and
assume the developing map takes fibers of the bundle R? {n} to lines of the form
87
R? {y} where y ? Rn.
That said, we may form the quotient of the commutative square in Equation
4.4 by each R-action to obey the commutative square as in Equation 4.13. Recall
here the action of the fundamental group and holonomy are the induced actions as
defined by Equation 4.8 and Equation 4.10 respectively. Because the holonomy by
hypothesis has no translational part in the Rn component of R?Rn, this means the
holonomy acts purely linearly on Rn in the diagram defined by Equation 4.13.
Up until this point we have yet to assume the developing image dev?(R?N)
meets the invariant line y = 0. For if it does, this means that the image dev?(N)
contains 0 ? Rn.
Precompose dev? : N ?? Rn with the trivial bundle map p : R?N ?? N to
obtain the square below in Equation 5.13 that commutes for each [?] ? ?1(M).
R? [?]N R?N
dev??p dev??p (5.13)
Rn Rn
hol[?]
Recall the holonomy acts purely linearly on Rn as seen in Equation 5.12. Thus
it preserves the radiant vector field R = ?yi?/?yi. By Lemma 3.1 and Lemma 3.2,
we may lift R to an auxiliary ?1(M)-invariant vector field R on N via the local
diffeomorphism dev? : N ?? Rn where dev??R = R. While R is complete, we do
not at the moment have a means to guarantee that R is complete. To do so, we must
lift R to the universal cover R?N , through the trivial bundle p : R?N ?? N .
As ?1(M) acts on R?N both properly and freely, and the projection R?N ??
88
N is ?1(M)-equivariant, Lemma 5.1 provides a ?1(M)-invariant vector field on R?N
that lifts the ?1(M)-invariant R on N . Denote this vector field by R?.
By similar arguments to Section 3.2, R? descends to a vector field on M which
is by compactness complete, and consequentially has a flow defined for all time. By
Lemma 3.3, the corresponding flow on R ? N is also complete. The remark after
Lemma 3.3 guarantees R is complete.
As R is a complete lift of the radiant vector field R on Rn, and 0 ? dev?(N),
this means by Lemma 3.5 that dev? : N ?? Rn may be promoted from a local
diffeomorphism to a global diffeomorphism.
Saturating N by the R-action induced by the parallel flow by means of Theo-
rem 4.1 yields that dev? : R ? N ?? R ? Rn is a diffeomorphism. Thus the affine
structure on M is complete and M is diffeomorphic to the quotient of R?Rn by a
subgroup of G as defined in Equation 5.12. This though contradicts Theorem 5.1
as the holonomy is by hypothesis reducible.
From this we obtain two immediate corollaries regarding Theorem 5.3.
Corollary 5.1. Let M be an (n+ 1)-dimensional closed affine manifold with n ? 1
whose holonomy admits an invariant line. If the holonomy acts on the invariant line
by translations and reflections, then the developing image cannot meet this invariant
line.
Proof. Let M be such a manifold as in the hypothesis. This means the holonomy
89
lies in the group
?????? ?? ??
?
??
?
?????? ?
?
?1 v t ??
H = ???? ?? ?? t ? R, vT ? Rn, and A ? GL(n,R)?0 A 0 ?? (5.14)
which is a finite extension of the group G as defined in Equation 5.12. Thus we may
lift to the double cover manifold C ??M to assume the holonomy lies inside G in
Equation 5.12. By Theorem 5.3, the developing map of C must miss the invariant
line of the holonomy. Since the developing maps of M and C are identical by Lemma
2.2, the developing map of M misses the invariant line as claimed.
As a final Corollary to Theorem 5.3, we have the following result.
Corollary 5.2. Let M be an (n+ k)-dimensional closed affine manifold with k ? 1
whose holonomy admits an invariant k-plane. If the holonomy acts on this invari-
ant k-plane by reflections and translations, the developing image cannot meet the
invariant k-plane.
Proof. Assume the holonomy acts on the invariant k-plane by translations alone so
the holonomy elements are of the form in Equation 4.17. By the result of Theorem
4.2, there exists k-complete parallel flows on M? defined by the lifts of the invariant
vector fields ?/?x1, ?/?x2, . . . , ?/?xk. Thus we may assume the developing pair
obeys the commutative diagram as found in Equation 4.16. One quotients the
diagram by the Rk-action to obtain the commutative square as in Equation 4.18
then apply the same arguments as done in the proof of Theorem 5.3 to show that
dev?(N) avoids 0 ? Rn, for otherwise one has that dev? : Rk ? N ?? Rk ? Rn
90
is a diffeomorphism, and thus M is complete with reducible holonomy violating
Theorem 5.1.
With this result established the case where the holonomy lies in the finite ex-
tension by reflections follows immediately from the argument provided in Corollary
5.2.
91
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