ABSTRACT Title of Thesis: SYNGE?S TRICK REVISITED Nicol as Virgilio Flores Castillo, Master of Arts, 2008 Thesis directed by: Professor Karsten Grove Department of Mathematics We review Synge?s trick and present some of its applications to Riemannian Geometry. We use it to prove Frankel, Weinstein-Synge and Wilking?s theorems, which concern manifolds of positive sectional curvature. First, we give the classical proofs of these theorems and then we present a reformulation of Synge?s trick in terms of a lower bound for the index of a special kind of geodesics. SYNGE?S TRICK REVISITED by Nicol as Virgilio Flores Castillo Thesis submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial ful llment of the requirements for the degree of Master of Arts 2008 Advisory Committee: Professor Karsten Grove, Chair/Advisor Professor Giovanni Forni Professor Elmar H. Winkelnkemper c Copyright by Nicol as Virgilio Flores Castillo 2008 A la memoria de mi padre. ii ACKNOWLEDGEMENTS I would like to thank my advisor, Prof. Karsten Grove, for suggesting me this interesting problem and giving me advice. I also would like to thank to my committee members: Prof. Giovanni Forni and Prof. Elmar H. Winkelnkemper. I would like to thank Cecilia Gonz alez Tok- man for useful discussions. I would like to thank professor Sergey Novikov for the valuable knowledge he shared in his courses. I also would like to thank my family for their support. The author is thankful to CONACYT, Mexico for supporting this re- search. iii TABLE OF CONTENTS List of Figures v 1 Introduction 1 2 Synge?s Trick and Some Theorems About Positive Sectional Cur- vature 3 2.1 First and Second Variational Formulas of the Energy Function . . . 3 2.2 Synge?s Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Classical Theorems: Frankel and Weinstein-Synge?s theorems . . . . 8 3 Index and Connectivity 18 3.1 Lower Bound for the Index of non-trivial N-geodesics . . . . . . . . 18 3.1.1 Reformulation of the Classical Theorems as an Index of N- geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.2 Direct proofs of the Classical Theorems . . . . . . . . . . . . 21 3.2 Wilking?s theorem: The Connectedness Principle . . . . . . . . . . . 23 3.3 Optimal General Index Theorem . . . . . . . . . . . . . . . . . . . 25 Bibliography 32 iv LIST OF FIGURES 2.1 Synge?s trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Frankel?s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Weinstein-Synge?s theorem . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Antipodal map on S2 . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1 Identi cation MIN !(M M)I0N 4 . . . . . . . . . . . . . . . . . 26 3.2 Identi cation MIN !(M M)I0N 4 . . . . . . . . . . . . . . . . . 27 3.3 Parallel vector elds ^P along ^ . . . . . . . . . . . . . . . . . . . . 29 v Chapter 1 Introduction In the present work, we will study some theorems about manifolds of positive curvature, all of which use a common idea introduced by Synge in [10]. As in [11], we call this idea Synge?s trick. Among these theorems are Frankel?s theorem, Weinstein?s theorem and Wilking?s theorem. This work was inspired by some ideas of my advisor, Prof. Grove, and also by [9]. First, we see the classical proofs of Frankel and Weinstein-Synge?s theorems, which are proved by contradiction. Then, we present the same theorems but now being reformulated as a computation of a lower bound for the index of a special kinds of geodesics, called N-geodesics. Then, we give a direct proof of Frankel and Weinstein?s theorems. We also present Wilking?s theorem, which also uses Synge?s trick to nd a lower bound for these special kind of geodesics. At the end, we formulate and prove an optimal general index theorem which encompass the theorems mentioned above. In the rst part of the thesis, we present the rst and the second variational formulas of the energy function 1 of paths. We also state and give proofs of the 1The energy function we use in this work, it is known in Physics as the action functional. 1 theorems mentioned in the above paragraph and illustrate how Synge?s trick in conjunction with the second variational formula for the energy function of paths is used in each of them. In the second part of the thesis, we explain how these theorems have really to do with the index of geodesics in a manifold M. 2 Chapter 2 Synge?s Trick and Some Theorems About Positive Sectional Curvature 2.1 First and Second Variational Formulas of the Energy Function In this section, we compute the rst and second variational formulas of the energy function, we proceed similarly as in [8] and [2]. Let (M;<>) be a Riemannian manifold and let p;q 2 M be two points of M. Let : [0;1] ! M be a curve that joins p with q; i.e. (0) = p and (1) = q. Consider a di erentiable variation f(s;t) of the curve . That is, f is a di erentiable mapping f : ( ";") [0;1]!M (s;t) 7!f(s;t) such that f(0;t) = (t). Remark 1. For t =constant, the curves ft(s) := f(s;t) : ( ";")!M are called the transversal curves of the variation. 3 Remark 2. Note that the vector eld @f@t(0;t) is the velocity vector eld 0(t) along the curve (t). Remark 3. The vector eld W(t) := @f@s(0;t) is called the variational vector eld along (t). Next, we will compute the rst variational formula of the energy function E(0) := E(ft(0)) := 12 Z 1 0 k 0(t)k2dt; where k 0(t)k2 =< 0(t); 0(t) >. Theorem 2.1.1 (First Variational Formula of the Energy Function). Let f(s;t) be a di erentiable variation of the curve , then E0(0) = ddsE(ft(0)) = W(1); 0(1) W(0); 0(0) Z 1 0 W(t);D dt 0(t) dt: Proof. By the properties of the Levi-Civita connection, we have that the following identities are satis ed for any parametrized surface f in a Riemannian manifold d ds @f @t(s;t); @f @t(s;t) = 2 D ds @f @t(s;t); @f @t(s;t) D ds @f @t = D dt @f @s where we have used the compatibility of the metric in the rst equation and the symmetry of the metric in the second equation. Then E0(s) = ddsE(ft(s)) =12 Z 1 0 d ds @f @t(s;t); @f @t(s;t) dt =12 Z 1 0 2 Dds@f@t(s;t);@f@t(s;t) dt = Z 1 0 D dt @f @s(s;t); @f @t(s;t) dt: 4 Using again the compatibility of the connection with the metric we have that d dt @f @s(s;t); @f @t(s;t) = D dt @f @s(s;t); @f @t(s;t) + @f @s(s;t); D dt @f @t(s;t) : Then E0(s) = Z 1 0 d dt @f @s(s;t); @f @t(s;t) dt Z 1 0 @f @s(s;t); D dt @f @t(s;t) dt = @f@s(s;t);@f@t(s;t) t=1t=0 Z 1 0 @f @s(s;t); D dt @f @t(s;t) dt = @f@s(s;1);@f@t(s;1) @f@s(s;0);@f@t(s;0) Z 1 0 @f @s(s;t); D dt @f @t(s;t) dt: So, at s = 0 we have that E0(0) = @f@s(0;1);@f@t(0;1) @f@s(0;0);@f@t(0;0) Z 1 0 @f @s(0;t); D dt @f @t(0;t) dt = W(1); 0(1) W(0); 0(0) Z 1 0 W(t);D dt 0(t) dt: Now, let us compute the second variational formula of the energy function. Let : [0;1] ! M be a geodesic that joins the points p and q of M(i.e. a critical point of the energy function E : (M;p;q)!R.) Let W1;W2 2 T be two vector elds along the curve . Let f be a 2- parameter di erentiable variation of ; that is, a di erentiable function f : U [0;1]!R (s1;s2;t)7!f(s1;s2;t) where U is a neighborhood of (0;0)2R2 in such a way that f(0;0;t) = (t); @f@s 1 (0;0;t) = W1(t) and @f@s 2 (0;0;t) = W2(t): 5 Then as in [8], the Hessian E (W1;W2) is de ned by E (W1;W2) := @ 2 @s2@s1E(ft(s1;s2)) (0;0): Theorem 2.1.2 (Second Variational Formula of the Energy Function). Let : [0;1] !M be a geodesic that joins the points p and q of M. Let f be a 2-parameter variation of as above. Then E (W1;W2) = @ 2E @s2@s1(0;0) = D ds2W1(1); 0(1) + W1(1);D dtW2(1) Dds 2 W1(0); 0(0) W1(0);DdtW2(0) Z 1 0 W 1(t); D2 dt2W2(t) +R 0(t);W 2(t) 0(t) : Proof. By the rst variational formula of the energy, we have that @ @s1E(ft(s1;s2)) = @f @s1(s1;s2;1); @f @t(s1;s2;1) @f @s1(s1;s2;0); @f @t(s1;s2;0) Z 1 0 @f @s1(s1;s2;t); D dt @f @t(s1;s2;t) dt: Then @E @s2@s1(s1;s2) = @ @s2 n @f @s1(s1;s2;1); @f @t(s1;s2;1) @f @s1(s1;s2;0); @f @t(s1;s2;0) Z 1 0 @f @s1(s1;s2;t); D dt @f @t(s1;s2;t) dto = Dds 2 @f @s1(s1;s2;1); @f @t(s1;s2;1) + @f @s1(s1;s2;1); D ds2 @f @t(s1;s2;1) Dds 2 @f @s1(s1;s2;0); @f @t(s1;s2;0) @f @s1(s1;s2;0); D ds2 @f @t(s1;s2;0) Z 1 0 D ds2 @f @s1(s1;s2;t); D dt @f @t(s1;s2;t) dt Z 1 0 @f @s1(s1;s2;t); D ds2 D dt @f @t(s1;s2;t) dt: Now, evaluate the above expression at (s1;s2) = (0;0). Since (t) = f(0;0;t) is a geodesic (i.e. Ddt@f@t(0;0;t) = 0) then we have that the fth term of the above 6 equality is zero. Also, if we interchange the order of the derivatives of the second and fourth terms and using the fact that for any parametrized surface f R @f @t(0;0;t); @f @s2(0;0;t) @f @t(0;0;t) = D ds2 D dt @f @t(0;0;t) D dt D ds2 @f @t(0;0;t) in the last term of the above equality, we get precisely the second varational formula of the energy function. 2.2 Synge?s Trick In the study of manifolds with positive sectional curvature there are several im- portant results that make use of Synge?s trick. Among these results are: Frankel?s theorem, Weinstein-Synge?s theorem and Wilking?s theorem. Although these theorems have di erent hypothesis, Synge?s trick can be used once we have a basic setting: given a complete connected Riemannian manifold M with positive sectional curvature and two distinct points p and q in M, suppose we have a minimal geodesic (t) that joins p and q; then choose a unit parallel vector eld W1(t) along (t), which is orthogonal to the tangent eld of the curve (t); then make a 1-parameter variation f(s;t) = f(s1 + s2;t) = f(s1;s2;t) of with variational eld @f@s(s;t)js=0 = @f@si(0;0;t) = W1(t) for i = 1;2 (in which the transversal curves (s) := f(s;0) and (s) := f(s;1) to are geodesics); then by 7 the second variational formula of the energy function of curves we get that E (W1;W1) = DdsW1(1); 0(1) + W1(1);DdtW1(1) DdsW1(0); 0(0) W1(0);DdtW1(0) Z 1 0 W 1(t); D2 dt2W1(t) +R 0(t);W 1(t) 0(t) dt: Since the vector eld W1(t) is parallel, then DdtW1(t) = 0, so the second and fourth terms of the above expression are zero. Also, since the curves (s) and (s) are geodesics then Dds 0(s) = 0 = Dds 0(s), so in particular, D dsW1(0) = D ds 0(0) = 0 and D dsW1(1) = D ds 0(0) = 0; so the rst and third terms of the second variational formula of the energy are also zero. Since the manifold M has positive sectional curvature, the second variational formula of the energy becomes E (W1;W1) = Z 1 0 W 1(t);R 0(t);W 1(t) 0(t) dt = Z 1 0 sec( 0(t);W1(t))dt< 0; then one gets shorter curves fs(t) (that join the point (s) to (s)) close to the original geodesic (t). In di erent situations, this is going to give us a contradic- tion, as we will see in the following theorems. 2.3 Classical Theorems: Frankel and Weinstein- Synge?s theorems Let us see how Synge?s trick is used in the theorems mentioned above. The fol- lowing theorem was proved by Frankel in 1960 in [3]. 8 p q M (t) (s) (s) W1(t) Figure 2.1: Synge?s trick Theorem 2.3.1 (Frankel). Let Mm be a complete connected Riemannian manifold with positive sectional curvature and let Ur and Ws be compact totally geodesic submanifolds such that r+s m, then Ur and Ws have a non-empty intersection. U W p=c(0) q=c(l) TpU TqW c(t) V(t) (s) (s) Figure 2.2: Frankel?s theorem Proof. Let U and W be two totally geodesic submanifolds of M. Suppose that U and W does not intersect. Then, there exists a minimal geodesic c(t), with length 9 l, that joins U and W say at the points p = c(0)2U, q = c(l)2W. Since c(t) is a minimal geodesic joining U and W, then c(t) is orthogonal to U and W at c(0) and c(l), respectively. Let fW TqM be the linear subspace obtained by parallel transporting TpU along c(t) at the point q. Since TpU is orthogonal to c at p, then fW is also orthogonal to c(t) at q. Note that TqW;fW TqM are linear subspaces of TqM. Then dim(TqW\fW) dimTqW + dimfW (dimTqM 1) = s+r m+ 1 1: So, TqW\fW 6= ;. Then, there exists a parallel vector eld V(t) along c(t) such that V(0)2TpU and V(l)2TqW. Now, let f(s;t), s2 [ ";"], t2 [0;l] be a variation along the curve c(t), in which the transversal curves are geodesics and which has variational eld V(t). Note that since U and W are totally geodesic submanifolds of M then the geodesics (s) = f(s;0); (s) = f(s;l) lie entirely in U and W, respectively. Then D dsV(0) = D ds @f @s(s;0) s=0 = Dds 0(s)js=0 = 0 and D dsV(l) = D ds @f @s(s;l)js=0 = Dds 0(s)js=0 = 0: 10 Therefore, by the second variation formula of energy we get that d2E ds2 (0) = D dsV(l);c 0(l) + V(l);DdtV(l) D dsV(0);c 0(0) V(0);DdtV(0) Z l 0 V(t);R dc dt;V dc dt dt = Z l 0 sec dc dt;V dt< 0; so we get shorter curves fs(t) than c(t) (that join U with W), which contradicts that c(t) was minimal. Also Weinstein used Synge?s trick in [11], the following theorem is a corollary of a more general result mentioned in this paper (Weinstein proved his statement for conformal di eomorphisms not just isometries.) Theorem 2.3.2 (Weinstein). Let f : Mm !Mm be an isometry of a compact, oriented Riemannian manifold Mm with secM > 0. If f is orientation-preserving if m is even and orientation-reversing if m is odd, then Fix(f) =fx2M : f(x) = xg6=;. Proof. Suppose that q6= f(q) for all q2M. Let d(q;f(q)) be the distance between q2M and f(q)2M. Then d(q;f(q)) > 0 for all q2M. Since M is compact and d : M ! R+ is a continuous function given by q7 !d(q;f(q)), then there exists p2M such that d(p;f(p)) > 0 is a minimum value for d. Since M is complete, then there exists a minimal geodesic : [0;l] !M that joins p and f(p) (i.e. (0) = p and (l) = f(p)) and such that k 0(t)k= 1 for all 11 t2[0;l]: Let N and N0 be the orthogonal complement of 0 at p and f(p), respectively. Note that N is a subspace of TpM and N0 is a subspace of Tf(p)M. We claim that dfp(N) = N0, or equivalently that dfp( 0(0)) = 0(l): In fact, consider the curve f : [0;l] !M. Note that f is a geodesic. Let p0 = 0(t0), where t06= 0;l. Then, by the triangle inequality and since is a minimal geodesic, we have that d(p0;f(p0)) d(p0;f(p)) +d(f(p);f(p0)) = d(p0;f(p)) +d(p;p0) = d(p;f(p)); since p is a minimum value for d, the above inequality becomes equality, so the curve (f ), formed by juxtaposition of and f , is a geodesic, so 0(l) = ddt(f )(t) t=0 = dfp( 0(0)); as we claimed. Now, consider the linear automorphism Pf(p);p dfp : TpM !TpM; where Pf(p);p : Tf(p)M ! TpM is the parallel transport from (l) = f(p) to (0) = p along (t). Note that Pf(p);p dfp leaves N invariant, since Pf(p);p dfp( 0(0)) = Pf(p);p( 0(l)) = 0(0): Let A : N ! N be the restriction of Pf(p);p dfp to N. Since Pf(p);p is orientation-preserving and det(dfp) = ( 1)m, we have that det(A) = det(Pf(p);p dfp) = det(Pf(p);p)det(dfp) = ( 1)m: 12 Since A : Rm 1!Rm 1 is an orthogonal transformation with detA = ( 1)m, we have that A has a xed point v2N. N N' V(t) 0(0) 0(l) =dfp( 0(0)) v dfp(v) (t) (s) (s) Figure 2.3: Weinstein-Synge?s theorem Now, we have all the necessary ingredients to use Synge?s trick. Let V(t) be a parallel extension of the vector v2N along the curve (t) (i.e. V(0) = v2N, V(l) = dfp(v)2N0 and DVdt (t) = 0 for all t2[0;l].) Now, let h(s;t), s2 [ ";"], t2 [0;l], be a variation along the curve (t) in which the transversal curves are geodesics and which has variational eld V(t). For example, h(s;t) can be constructed in the following way: let (s), s2[ ";"], be a curve such that (0) = p and 0(0) = v2N. Then, the curve (s) := f (s) is such that (0) = f( (0)) = f(p) and 0(0) = dfp( 0(0)) = dfp(v)2N0. Let h(s;t) = exp (t)(sV(t)); s2[ ";"]; t2[0;l]: Then h(s;0) = exp (0)(sV(0)) = expp(sv) = (s); h(s;l) = exp (l)(sV(l)) = expf(p)(sdfp(v)) = (s) 13 and @ @sh(s;t) s=0 = exp (t) 0 (V(t)) = IdT (t)M(V(t)) = V(t): So V(t) is the variational eld of h(s;t) with DVdt = 0. Also, since and are geodesics, then D dsV(0) = D ds @h @s(s;0)js=0 = Dds 0(0) = 0 and D dsV(l) = D ds @h @s(s;l)js=0 = Dds 0(0) = 0: Then, by the second variation formula of the energy and since M has positive sectional curvature we get that d2E ds2 (0) = D dsV(l); 0(l) + V(l);DdtV(l) D dsV(0); 0(0) V(0);DdtV(0) Z l 0 V(t);D 2V dt2 +R d dt;V d dt dt = Z l 0 V(t);R d dt;V d dt dt = Z l 0 sec d dt;V dt< 0; so we get shorter curves hs(t) (which join points q2M to f(q) where q6= p) than (t), which contradicts that d(p;f(p)) was a minimum. The hypothesis about f : M2m !M2m being orientation preserving can not be relaxed, as it is shown in the following 14 Example 1. Let M = S2 and f = id be the antipodal map of S2 id : S2 !S2 x7 ! x: Then f is orientation-reversing since degf = 1, also note that id x : TxS2 !TxS2 v7 ! v and clearly id does not have xed points. p -v -x v x CASE 1 p v -v x-x CASE 2 S2 S2 Figure 2.4: Antipodal map on S2 Similarly, the hypothesis aboutf : M2m+1!M2m+1 being orientation-reversing can not be relaxed as it shows the following Example 2. Let M = S3 and f = id be the antipodal map of S3. Then f is orientation-preserving, since degf = 1, and f does not have x points. 15 As a corollary of Weinstein?s theorem we have a weaker version proved earlier by Synge. I borrowed the main ideas of the proof from [1], [11] and [7]. Corollary 2.3.1 (Synge). Let Mm be a compact, connected, Riemannian manifold with positive sectional curvature. Then, a) If m is even and (1) Mm is orientable then 1(M) = 1. (2) Mm is not orientable then 1(M) = Z2. b) If m is odd, then Mm is orientable. Proof. (a1) Suppose m is even and Mm is orientable. Let p : fM !M be the universal cover of M. Let fM have the covering metric (i.e. the pull-back metric of M) and let fM be oriented in such a way that p preserves the orientation. Since fM satis es the same curvarture conditions as M, then fM has positive sectional curvature. Then, by Myers? theorem, fM is compact. Let D : fM ! fM be a Deck transformation of M. Then, by the way we oriented fM, D is an orientation-preserving isometry. Since m is even, by Weinstein?s theorem, we have that D has a xed point, and therefore D is the identity map of fM. Since 1(fM) = 1, then the group of Deck transformations G can be identi ed with 1(M). Then 1(M) = G =f1g. (a2) Suppose m is even and Mm is not orientable. Let fM be the orientable double cover of M and let fM have the covering metric. As in part (a1), since fM 16 satis es the same curvature conditions as M, then fM has positive sectional curvature. Then, by Myers? theorem, fM is compact. Since m is even and fM is orientable, by (a1) we have that 1(fM) = 1. Then, since fM is a double cover of M we have thatj 1(M)j= 2, then 1(M) = Z2. (b) Suppose m is odd and M is not orientable. Let fM be the orientable double cover ofM. Let fM have the covering metric. As in part (a) fM is also compact (by Myers? theorem or being the double cover of a compact manifold.) Since fM is a double cover of M, we have that [ 1(M) : p#( 1(fM))] = 2, so p#( 1(fM)) E 1(M). So the covering p : fM !M is regular. Then, the group of Deck transformations G = 1(M)=p#( 1(fM)) = Z2. Let D 2 G and such that D 6= id. Then, D is an orientation-reversing isometry of fM. Since m is odd, by Weinstein?s theorem, D has a xed point, but this contradicts that D6= id. 17 Chapter 3 Index and Connectivity 3.1 Lower Bound for the Index of non-trivial N- geodesics All the previous theorems can be studied considering the index of N-geodesics in M introduced in [5]. That is, let Mm be a complete, connected Riemannian manifold with positive sectional curvature. Let Nn ,! Mm Mm be a closed totally geodesic submanifold. Let MIN = f 2C0(I;M) : ( (0); (1)) 2Ng; i.e. MIN is the space of curves (t) : [0;1] !M such that ( (0); (1))2N. As in [5] we say that the geodesic in M is an N-geodesic if 2MIN and ( 0(0); 0(1)) is normal to N, (3.1) where 0(t) denotes the velocity vector of at t and M M has the product metric. Observe that T MIN =fX 2T C0(I;M) : (X (0);X (1))2T( (0); (1))Ng: From [4], we get the following remarks. 18 Remark 4. When N =4 M M is the diagonal submanifold of M M, we have that N-geodesics correspond just to closed geodesics in M. Remark 5. When N = U W, where U and W are submanifolds of M, an N- geodesic in M is just a geodesic in M that starts orthogonal to U and ends orthogonal to W. Then T MIN =fX 2T C0(I;M) : (X (0);X (1))2T( (0); (1))Ng =fX 2T C0(I;M) : (X (0);X (1))2T (0)U T (1)Wg: Remark 6. If f : M !M is an isometry on M and N = Graph(f), then an N-geodesic corresponds to an f-invariant geodesic; i.e. a geodesic in M with the property df (0)( 0(0)) = 0(1). Also, since T( (0); (1))Graph(f) =f(v;df (0)(v)) : v2T (0)M)g; we have that T MIN =fX 2C0(I;M) : (X (0);X (1))2T( (0); (1))Graph(f)g =fX 2C0(I;M) : (X (0);df (0)(X (0)))2T (0)M df (0)(T (0)M)g: So, in particular 02T MIN. 3.1.1 Reformulation of the Classical Theorems as an Index of N-geodesics Now, let us reformulate the idea of nding shorter curves in the case of Frankel and Weinstein theorems as an index of non-trivial N-geodesics. 19 Theorem 3.1.1 (Frankel). Let Mm be a complete connected Riemannian manifold with positive sectional curvature. Let N = Ur Ws M M where Ur and Ws are compact totally geodesic submanifolds of M such that r+s m, then the index of non-trivial N-geodesics is s+r m+ 1. Proof. Let (t) : [0;1] ! M be a non-trivial N-geodesic. Then by the above remark, we have that is a geodesic that joins U to W starting orthogonal to U and ending orthogonal to W. Using a similar procedure as in the original proof of Frankel?s theorem, we can nd a parallel vector eld V(t) along the curve (t), which is orthogonal to this curve, so V(0) 2T (0)U and V(1) 2T (1)W; i.e V 2T MIN. Then, performing a variation of that has variational eld V(t), in which the transversal curves are geodesics, using the second varational formula E : T MIN T MIN ! R of the energy function and the fact that M has positive sectional curvature we obtain that E (V;V) = Z 1 0 sec( 0;V)dt< 0; so the index of is s+r m+ 1. Theorem 3.1.2 (Weinstein). Let f : Mm !Mm be an isometry of a compact, oriented Riemannian manifold Mm with secM > 0. Let N = Graph(f). If f is orientation-preserving if m is even and orientation-reversing if m is odd, then the index of non-trivial N-geodesics is 1. Proof. Let (t) : [0;1] ! M be a non-trivial N-geodesic. Then, by the above remark is an f-invariant geodesic. Then, using an analogous procedure as in the original proof of Weinstein?s theorem, we nd a parallel vector eld V(t) along (t) 20 which is orthogonal to this curve such that V(1) = dfp(V(0)) (note that V 6= 0). Then, by the above remark we have that V 2T MIN. Then, making a variation of that has variational vector eld V(t), using the second variational formula E : T MIN T MIN !R of the energy function and the fact that M has positive sectional curvature we obtain that E (V;V) = Z 1 0 sec( 0;V)dt< 0; so the index of is at least 1. The hypothesis of f : Mm!Mm about being orientation-preserving (revers- ing) according to m is even (odd) for the computations of the index of N-geodesics can not be relaxed. The same examples given in the rst section illustrate this fact. Example 3. Let M = S2 and f = id be the antipodal map of S2. Then f is an orientation-reversing isometry. Let x2S2 and consider a minimal geodesic in S2 that joins x to x. Then is a non-trivial Graph(f)-geodesic; however, it has index 0. Example 4. Let M = S3 and f = id be the antipodal map of S3. Then f is an orientation-preserving isometry. Let x2S3 and consider a minimal geodesic in S3 that joins x to x. Then is a non-trivial Graph(f)-geodesic; however, it has index 0. 3.1.2 Direct proofs of the Classical Theorems Now, let us focus our attention to the energy function for paths restricted to the space MIN. As in [4], the critical points of the energy function restricted to MIN 21 correspond to N-geodesics. This is true, since if is a critical point for the energy function restricted to MIN, then E0(0) = 0. The fact that is a geodesic is a consequence of proposition 1.5 in [4]. Then, by the rst variational formula of the energy function, we have that 0 = E0(0) =hW(1); 0(1)i hW(0); 0(0)i =h(W(1);W(0));( 0(0); 0(1))iM M for any W(t)2T MIN. The above equality is precisely condition (1) of the de ni- tion of an N-geodesic; that is, ( 0(0); 0(1))2(T( (0); (1))N)?: That an N-geodesic is a critical point of the energy function restricted to MIN is clear if we use the facts that is a geodesic and satis es condition (3.1) in the rst variational formula of the energy function. Remark 7. Grove and Halperin proved in [6] that the energy function E : MIN ! R satis es condition (C)of Palais and Smale (a necessary condition for making critical point theory, like Morse theory on in nite dimensional manifolds) i the function dist e : N ! R is proper, where dist : M M ! R is the distance function in M and e : N ,!M M is the inclusion map. With the computations of a lower bound for the index of non-trivialN-geodesics in the cases discussed above, we can now give a direct proof to Frankel?s and Weinstein?s theorems. Theorem 3.1.3 (Frankel). Let Mm be a complete connected Riemannian manifold with positive sectional curvature and let Ur and Ws be compact totally geodesic submanifolds such that r+s m, then Ur and Ws have a non-empty intersection. 22 Proof. Let N = U W. Let be a non-trivial N-geodesic. So, is a critical point of the energy function (restricted to MIN). By our index estimates, we have that the index of s + r m + 1. Then is not a minimum for the energy function, otherwise it would have index 0. Then, the minimum is reached1 at trivial N-geodesics, i.e. points of N = U W. This means that U must intersect W. Theorem 3.1.4 (Weinstein). Let f : Mm !Mm be an isometry of a compact, oriented Riemannian manifold Mm with secM > 0. If f is orientation-preserving if m is even and orientation-reversing if m is odd, then Fix(f) =fx2M : f(x) = xg6=;. Proof. Let N = Graph(f). Let be a non-trivial N-geodesic. So, is a critical point of the energy function (restricted to MIN). By our index estimates, we have that the index of 1. Then is not a minimum for the energy function, otherwise it would have index 0. Then, the minimum is reached2 at trivial N- geodesics, i.e. points of N = Graph(f) = f(p;f(p)) : p2Mg. This means that there exists a point x2M such that x = f(x). 3.2 Wilking?s theorem: The Connectedness Prin- ciple The next theorem, which uses Synge?s trick and Morse Theory, was proved by Wilking in [12]. It is in some sense, a special case of Frankel?s theorem where 1see remark 7 2see remark 7 23 N = U U M M and U is a compact totally geodesic submanifold of M. It uses Synge?s trick similarly as in the case of Frankel?s theorem to compute a lower bound for the index of N-geodesics. Theorem 3.2.1 (Connectedness Principle). Let Mm be a compact manifold with positive sectional curvature, suppose that Um k Mm is a compact totally geodesic embedded submanifold of codimension k. Then the inclusion map Um k ,!Mm is (m-2k+1)-connected. Proof. Let N = U U M M. Let MIN = MIU U be the space of smooth curves (t) : [0;1] !M such that ( (0); (1))2U U; i.e. the space of smooth curves in M that start and end in U. Consider the energy function de ned on MIN = MIU U by E( ) = 12 Z 1 0 k 0(t)k2dt: Since Um k can be embedded in MIU U as the set of point paths, we have that E 1(0) = Un k. We will use Morse Theory to show that the space MIU U has the homotopy type of a CW-complex which consists of cells of dimension m 2k+1 attached to the 0 skeleton Um k. By the rst variational formula for the energy of paths, we have that the critical points of E are precisely the geodesics of M that start and end perpendicular to U; i.e. (U U)-geodesics. Let : [0;1] !M be one of such geodesics. We will use the same idea as in the proof of Frankel?s theorem, to compute a lower bound for the index of (t). Let W1 = T (0)U T (0)M be the tangent space to U at the point (0) and let W2 T (0)M be the linear subspace obtained by parallel transporting T (1)U along (t) at the point (0). Since T (1)U is orthogonal to 24 at (1), then W2 is also orthogonal to at (0). Then dim(W1\W2) dimW1+dimW2 (dimM 1) = 2(m k) (m 1) = m 2k+1: Then, there exist m 2k+1 parallel elds V(t) along (t) such that V(0)2T (0)U and V(1)2T (1)U. Similarly as in the proof of Frankel?s theorem, for every parallel eld V(t) we can construct a variation of with variational eld V(t) in which the transversal curves are geodesics, so using the second variational formula for the energy and the fact that secM > 0, we have that d2Eds2 (0) < 0. Then, the index of is m 2k + 1. Now using Morse Theory, see [8] section 22, we have that there is a Morse function E0 that is C1 close to E, such that E0 = E on a neighborhood of U = E 1(0), and has index m 2k + 1 on any geodesic 2 MIU UnU. Then r(MIU U;U) = 0 for 0 >< >>: 1(t) if t2[0; 12]; 2(1 t) if t2[12;1]: 1(t) 2(1 t) (t) M 4 N M M ^ (t) = ( 1(t); 2(t)) Figure 3.2: Identi cation MIN !(M M)I0N 4 Remark 8. N-geodesics are in 1-1 correspondence with (N 4)-geodesics un- der the identi cation given above. This can be veri ed either by checking that Ej0MI N (0) = Ej0(M M)I N 4 (0) or by using directly the de nition of a curve being an N-geodesic and (N 4)-geodesic, respectively and proving inclusion of sets. Let 2 MIN be a non-trivial N-geodesic. Let ^ 2 (M M)I0N 4 be the corresponding (N 4)-geodesic. That is, ^ (t) = ( 1(t); 2(t)) = ( (t); (1 t)) t2I0: Then since ^ is an (N 4)-geodesic, we have by the remark (5) that is a geodesic that starts orthogonal to N and ends orthogonal to 4. Let ^P(t) be a vector eld along ^ such that ^P(0)2T^ (0)N and ^P(1)2T^ (1)4; that is, ^P 2T^ (M M)I0N 4. Then ^P corresponds to a vector eld P along given by 27 P(t) = 8 >>< >>: P1(t) if t2[0; 12]; P2(1 t) if t2[12;1] such that (P(0);P(1))2T( (0); (1))N; i.e. P 2T MIN: Indeed, P(t) is well-de ned since the condition ^P(1=2) = (P1(1=2);P2(1=2))2T^ (1 2) 4= T( (1 2); ( 1 2)) 4 means that P1(12) = P2(12). Since ^P(0) = (P1(0);P2(0)) = (P(0);P(1)) and ^ (0) = ( 1(0); 2(0)) = ( (0); (1)), then the condition ^P(0) 2 T^ (0)N is a di erent way to write (P(0);P(1)) 2 T( (0); (1))N. Therefore, ^P 2T^ (M M)I0N 4 if and only if P 2T MIN. Moreover, it is clear that ^P 2T^ (M M)I0N 4 is a parallel vector eld along ^ if and only if P 2T MIN is a parallel vector eld along . Now, let us compute a lower bound for the index of a non-trivial N geodesic (which is the same that the index of the corresponding non-trivial (N 4)- geodesic.) Using the same idea as in the classical proof of Frankel?s theorem the number of parallel elds ^P along a non-trivial (N 4)-geodesic ^ which are orthogonal to the velocity vector eld of the curve ^ is at least dimN + dim4 (dim(M M) 1) = n+m (2m 1) = n m+ 1: 28 4 N M M ^ (t) ^P(t) Figure 3.3: Parallel vector elds ^P along ^ such that ^P ?^ 0 Note that the condition ^P ?^ 0 just means for P and 0 that 0 =h(P1(t);P2(t));( 01(t); 02(t))iM M 8t2I0 =h(P(t);P(1 t));( 0(t); 0(1 t))iM M 8t2I0 =h(P(t); 0(t))iM hP(1 t); 0(1 t)iM 8t2I0 which happens if and only if h(P(t); 0(t))iM =hP(1 t); 0(1 t)iM 8t2I0: (3.2) So, there are at least n m + 1 candidates P(t) in which the Hessian of the energy function EjMI N is going to be negative-de nite. Let P =fP 2T MIN : P is parallel along and satis es (3:2)g. Then dimP n m+ 1: So we get the following 29 Theorem 3.3.1 (Lower bound for the index of non-trivial N-geodesics). Let Mm be a complete, connected Riemannian manifold that has positive sectional curva- ture. Let Nn Mm Mm be a closed totally geodesic submanifold of M M. Then, the index of a non-trivial N-geodesic is at least (1) n m if 02P. (2) n m+ 1 if 062P. Proof. Suppose W 2P and W 62span( 0), then by the second variational formula of energy we get E (W;W) = D dsW(1); 0(1) + W(1);DdtW(1) D dsW(0); 0(0) W(0);DdtW(0) Z 1 0 W(t);D 2 dt2W(t) +R( 0(t);W(t)) 0(t) dt = Z 1 0 hW(t);R( 0(t);W(t)) 0(t)idt = Z 1 0 sec(M)dt< 0; where we have used in the second equality the fact that W(t) is parallel along (i.e. DdtW(t) = 0 for all t2I) and also we have performed a variation h(s;t) with variational vector eld W(t) in which the transversal curves h(s;0) and h(s;1) are geodesics, so DdsW(1) = 0 = DdsW(0) (this last step can be done since N M M is totally geodesic). Now, suppose 02P. Then, by the same facts as in the above formula, the rst four terms in the second variational formula of energy are zero, then we have that E ( 0; 0) = Z 1 0 h 0(t);R( 0(t); 0(t)) 0(t)idt = 0; 30 since R( 0(t); 0(t)) 0(t) = 0 for all t2I. Therefore, the index of a non-trivial N geodesic is at least n m if 02P and if 062P then the index is at least n m+ 1. Remark 9. Note that since is a geodesic, then 0 is parallel and satis es trivially the condition (3.2) above. Then 02P whether or not 02T MIN =fX 2T C0(I;M) : (X (0);X (1))2T( (0); (1))Ng; that is, whether or not ( 0(0); 0(1))2T( (0); (1))N. So, in the case N = Ur Vs with Ur;Vs M compact totally geodesic submanifolds of Mm, we obtain using the theorem above that the index of non- trivial N-geodesics is at least r + s m + 1, since in this case ( 0(0); 0(1)) 62 T( (0); (1))N (since ( 0(0); 0(1)) 2 (T( (0); (1))N)?). So, as in Frankel?s theorem, when r +s m then the index of is at least 1. However, in the case N = Graph(f), where f : M ! M is an isometry, we get using the above theorem that the index of a non-trivial N-geodesic is at least n m = m m = 0, since in this case 0 2P (since ( 0(0);f ( 0(0))) = ( 0(0); 0(1)) 2T( (0); (1))N). So, in this case we do not get anything we already knew. So, if we assume that Mm is compact, the lower bound for a non-trivial N-geodesic we obtain using Weinstein-Synge?s theorem (i.e. at least 1) is stronger than the lower bound we obtain using the above theorem, the reason is that the theorem for the lower bound for non-trivial N-geodesics neglects the hypothesis about f being orientation-preserving (reversing) in case m is even (odd). Precisely, examples (3) and (4) illustrates this fact. Notice that the bounds in this theorem are sharp: examples (3) and (4) satisfy part 1 of the theorem whereas Frankel?s theorem satis es part 2. 31 It is surprising that although Synge?s theorem was the rst one to be proved, it is actually more di cult to obtain than the other theorems (Frankel, Wilking) using the computation of the index on non-trivial N-geodesics. 32 BIBLIOGRAPHY [1] M. do Carmo. Riemannian Geometry. Birkhauser, 1992. [2] N. Flores Castillo. Una Aproximaci on Finita al Espacio de Caminos r. Universidad de Guanajuato-CIMAT (tesis de licenciatura), 2004. [3] T. Frankel. Manifolds of positive curvature. Paci c J. Math., 11:165{174, (1961). [4] K. Grove. Condition (C) for the energy integral on certain path spaces and applications to the theory of geodesics. J. Di . Geom., 8:207{223, (1973). [5] K. Grove. Geodesics Satisfying General Boundary Conditions. Comment. Math. Helv., pages 376{381, (1973). [6] K. Grove; S. Halperin. 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