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Toponogov's Theorem and Applications

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Toponogov's Theorem and Applications Top onogovs Theorem and Applications by Wolfgang Meyer These notes have been prepared for a series of lectures given at the College on Dierential Geometry at Trieste in the Fall of The lectures center around To p onogovs triangle comparison theorem critical p oint theory and applications In the short amount of time available not all the asp ects can be covered We fo cus on those applications which seem to be most imp ortant and at the same time most suitable for an exp osition Some basic knowledge in geometry will be assumed It has b een provided by K Grove in the rst series of these lectures Nevertheless we try to keep the lectures selfcontained and indep endent as much as p ossible For the result ab out the sum of Betti numb ers in section a lemma from algebraic top ology is needed A pro of for this result has b een provided in the app endix I am indebted to U Abresch for many helpful conversations and also for writing and typing the app endix Contents Review of notation and some to ols Covariant derivatives Jacobi elds Interpretation of curvature in terms of the distance function The levels ofadistance function Data in the constant curvature mo del spaces The Riccati comparison argument The Top onogov Theorem Applications of Top onogovs Theorem An estimate for the number of generators for the fundametal group Critical p oints of distance functions The diameter sphere theorem A critical p oint lemma and a niteness result An estimate for the sum of Betti numbers The soul theorem App endix A top ological Lemma Review of notation and some to ols Covariant derivatives We consider a complete Riemannian manifold M with tangent bundle TM and Rie mannian metric h i and corresp onding covariant derivative r of Levi Civita which ector elds is the unique torsion free connection for which h i is parallel ie for anyv X Y Z on M we have r Y r X X Y X Y and X hY Z i hr YZi hY r Z i X X The last two equations are equivalentto the Levi Civita equation hr YZi X hY Z i Y hZ X i Z hX Y i X hZ X Y i hY Z X i hX Y Z i If M is an arbitrary manifold and f M M a dierentiable map f T M TM denotes the dierential of f r naturally extends to a covariant derivative for vector elds along f For any vector eld A on M and any vector eld Y along f ie Y M TM satises Y f where TM M denotes the pro jection the covariant derivative r Y is well dened Due to the fact that r Y dep ends only A A p on A and the values of Y in a neighb ourho o d of the p oint p this extension is uniquely p determined by requiring the chain rule r X r X for any tangentvector vTM v f f v and any vector eld X on M In a similar way the corresp onding covariant derivative for tensor elds carries over to a covariant derivative for tensorelds along a map As a consequence one obtains for example the Cartan structural equations for the Levi Civita connection r f B r f A f A B A B R f A f B Y r r Y r r Y r Y A B B A AB where R is the curvature tensor of r A B are vector elds on M and Y is a vector eld along the map f For a curve c I M the parameter vector eld on I with resp ect to the parameter t will be denoted by or D c t c j is the the tangent vector of c at t The t t t t covariant derivative r Y for a vector eld Y along c is abbreviated by Y A parallel D t vector eld Y along c is characterized by the linear dierential equation Y a geo desic curve by the nonlinear second order equation c For consistency reasons we avoid the often found notation r c resp r Y for the expressions r c c c D t resp r Y when Y is a vector eld along c The inconsistency of such notation D t b ecomes apparent when c is a singular curve for example a constant curve and Y a nonconstant vector eld along c If X is a vector eld on M r X r X chain c D c t rule is well dened The exp onential map exp TM M is determined by the initial value problem for geo desics If vT M then exp v c where c is the geo desic with initial condition p The restriction of exp to the tangent space T M at p is denoted c p and c v p by exp Notice that for complete manifolds the exp onential map is dened on all of p TM by the HopfRinow theorem For a function f M IR and a vector eld X on M Xf denotes the derivative of f in direction X The gradientof f is dened via the equation hgrad fXi Xf and the Hessian Hess f of f by Hess f X r grad f X Hess f is a selfadjoint endomorphism eld ie hr grad fYi hr grad fXi X Y Imp ortant functions on a Riemannian manifold are distance functions or lo cal dis tance functions from some point in M or from a submanifold of M A lo cal distance function is a function in an op en subset U of M considered as a Riemannian subman ifold If p U M and r q dist p q r q dist q p then r q r q M U U U r may b e dierentiable in p oints where r fails to b e dierentiable Atypical example U arises as follows Let c M b e an injective geo desic segment with initial p oint p c and without conjugate p oints Then there is a neighborhood U of c where r is dierentiable However r is not dierentiable in any p oint of the cut lo cus U of p For explicit examples lo ok at geo desics on a cylinder On the set of p oints where a lo cal distance function is dierentiable it satises jjgrad f jj The gradient lines of any function with this prop erty are geo desics D E parametrized by arc length since r grad fX hHess f grad fXi grad f hHess fXgrad f i hr grad fgrad f i X hgrad fgrad f i for any vector eld X X on M and hence r grad f Therefore the level surfaces of such a function grad f are equidistant They are referred to as a family of parallel surfaces Jacobi elds Jacobi elds J along a geo desic arise naturally as variational vector elds in one pa rameter families of geo desic lines and are characterized by the linear second order dierential equation J R J c c If V is a geo desic variation of c ie V I M is dierentiable and j is a V t ct and t V t s is a geo desic for all s then J tV t s Jacobi eld along c j J t r r V D j r r V D j r V D D t D D s t D D t t D s t t t t s t z j r r V D j r r V D t D D t t D D t t s s t z R V D V D V D j R J c c s t t t t Therefore the Jacobi equation is the linearization of the geo desic equation along c Notice that V can b e written in the following way If p is the curve ps V s and Y the vector eld along p given by Y s V D j then V t s exp tY s The t s initial conditions of the Jacobi eld in terms of p and Y are J p J Y Y is the initial vector of the geo desic c Any tangentvector u to TM can b e written as the tangent vector u Y j of a curve s Y j TM Y is a vector eld along s the base curve ps Y j If Y and V are dened as ab ove we nd exp u s d exp Y j V D j J Therefore the dierential of the exp onential map is s completely determined by Jacobi elds J w along the For example the Jacobi eld with initial conditions J geo desic exp tv is obtained from the variation V t s exp tv sw Here ps is the constant curve Y s v sw J t exp j tw J exp j w This shows tv p v that the dierential of the restriction expj is determined by Jacobi elds on M T M p with these initial conditions Interpretation of curvature in terms of the distance func tion Consider two geo desics c c emanating from a p oint p in M c exp v c exp w v w T M and the distance L distc c in a neighb orho o d of p o zero Then the fourth order Taylor formula for L is given by L jjv w jj hR v ww v i O When v w this implies for hR v ww v i Ljjv w jj O jjv w jj For linearly indep endent vectors v w satisfying jjv jj jjw jj this can be rewritten as Ljjv w jj K v w hv wi O where K v w ist the sectional curvature of the plane spanned by v and w Therefore L grows faster than linear if K and slower than linear if K in a neighborhood of To prove we consider the variation c V t exp t exp c w w w p0 p0 p0 v v v K < 0 K ≡ 0 K > 0 Figure interpretation of sectional curvature c1 (ε) c1 T w (ε,t) p0 v aε (t) c0 E(ε,t) c0 (ε) Figure setup for the pro of of for small values of and t The parameter tangent elds along V are E V D and T V D The parameter curves a t V t are geo desics connecting the t points c and c T is the tangent eld of the geo desics and t E j is a Jacobi t eld along a and E j c E j c tjj is the length of a so that Notice that jja Ljja tjj jjT jj t which is constant in t for xed The derivatives of H L up to the fourth order are given by H hr T Tij D t D E H r T T hr T r T i j D D t D E D E D T j H r T T r T r t D D D D E D E IV H r T T hr T r T i r T r T j D t D D D D We will nowevaluate these derivatives at t in order to nd the co ecients for the T will be used frequently E and r T r Taylor formula The equation r D D D t t during this calculation Also notice that T j since V t p We have t E j E j r r D D since E j c and E j c From the Jacobi prop erty of E we obtain r r E R E T T D D t t so that r r E j D D t t t Hence t E j is a linear vectoreld along the constant curve a Since E j t c v E j c w it follows E j v tw v t With this information we can already evaluate H and H H
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