Geodesic flows in manifolds of nonpositive curvature Patrick Eberlein. Table of Contents

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1 1 Geodesic flows in manifolds of nonpositive curvature Patrick Eberlein Table of Contents I. Introduction - a quick historical survey of geodesic flows on negatively curved spaces. II. Preliminaries on Riemannian manifolds A. Riemannian metric and Riemannian volume element B. Levi Civita connection and covariant differentiation along curves C. Parallel translation of vectors along curves D. Curvature E. Geodesics and geodesic flow F. Riemannian exponential map and Jacobi vector fields G. Isometries and local isometries H. Geometry of the tangent bundle with the Sasaki metric III. Manifolds of nonpositive sectional curvature A. Definition of nonpositive curvature by triangle comparisons B. Growth of Jacobi vector fields C. The Riemannian exponential map is a covering map. Theorem of Cartan - Hadamard. D. Examples : Riemannian symmetric spaces E. Convexity properties and the Cartan Fixed Point Theorem F. Fundamental group of a nonpositively curved manifold. G. Rank of a nonpositively curved manifold IV. Sphere at infinity of a simply connected manifold of nonpositive sectional curvature A. Asymptotic geodesics and cone topology for M ~ ( ) B. Busemann functions and horospheres Supported in part by NSF Grant DMS

2 2 C. Extension of isometries to homeomorphisms of the sphere at infinity. D. Relating the action of the geodesic flow of M on T 1 M to the action of π 1 (M) on M ~ ( ) V. Measures on the sphere at infinity A. Harmonic measures {ν p : p M ~ } B. Patterson - Sullivan measures {µ p : p M ~ } C. Lebesgue measures {λ p : p M ~ } D. Barycenter map for probability measures. VI. Anosov foliations in the unit tangent bundle T 1 M A. Stable and unstable Jacobi vector fields. B. The stable and unstable foliations E s and E u in T(T 1 M) C. The strong stable and strong unstable foliations E ss and E uu in T(T 1 M). D. Conditions for the foliations E ss and E uu to be Anosov. VII. Some outstanding problems of geometry and dynamics A. The Katok entropy conjecture B. Smoothness of Anosov foliations and Riemannian symmetric spaces C. The geodesic conjugacy problem D. Harmonic and asymptotically harmonic spaces E. Early partial solutions. VIII. The work of Besson - Courtois - Gallot A. Statement of the main result. B. Corollaries of the main result. C. Sketch of the proof of the main result. IX. References I. Introduction In this section we give a very brief survey of geodesic flows on spaces of negative curvature. For a more complete account see the articles [Ano], [EHS] and [Hed] and the references in these articles. From the beginning, at least from the 1920 s, the properties of the geodesic flow on a space of strictly negative curvature have been studied with a variety of methods - geometry, analysis, ergodic theory and even brute force matrix manipulations that later

3 3 became more sophisticated Lie theoretic methods. This has made the subject of geodesic flows both more interesting and more difficult for researchers who may have to digest unfamiliar points of view. These notes are an attempt to describe some results and methods from Riemannian geometry that will be useful in understanding the geometric approach. It is useful to consider only unit speed geodesics in a Riemannian manifold M, which we always assume to be C, and in this case we obtain a flow called the geodesic flow on the space T 1 M of all unit vectors tangent to M. Given a unit vector v tangent to M and a real number t we define g t (v) to be the velocity at time t of the unique geodesic γ v with initial velocity v. If M is compact, or more generally geodesically complete, then g t : T 1 M T 1 M is defined for every real number t, and the flow transformations {g t } are easily seen to obey the rule g t+s = g t ο g s = g s ο g t for all real numbers s and t. If M has dimension n, then T 1 M has dimension 2n-1 since the natural projection π : T 1 M M has fibers that are unit spheres of dimension n-1, the unit vectors at a fixed point of M. In the 1920 s it was suspected that geodesic flows on the unit tangent bundle T 1 M of a compact 2-dimensional manifold with Gauss curvature K 1 should have very special properties : a dense set of vectors whose orbits under {g t } are periodic, a set of vectors of full measure whose {g t } orbits are dense in T 1 M and a variety of even stronger properties including ergodicity and mixing. The leading researchers during this period were Nielsen, Koebe, Morse and others, and a good survey may be found in [Hed]. The early results exploited the algebraic structure of SL(2, ), the group of 2 x 2 real matrices of determinant 1 and also, after equating elements A and A, the identity component of the group of isometries of the hyperbolic plane with Gaussian curvature 1. During the 1930 s G. Hedlund led the transition to more geometric and analytic methods, and he succeeded in proving ergodicity and mixing properties for the geodesic flow in T 1 M for a compact, 2-dimensional manifold with Gauss curvature K 1. During the period 1937 to 1942 E. Hopf [Ho 1, 2] succeeded in extending the new geometric methods to obtain ergodicity in the case where M is a compact, 2-dimensional manifold with strictly negative but nonconstant negative curvature. Hopf s method involved using what later came to be called the stable and unstable Anosov foliations in T 1 M. In the case that dim M = 2 these both are 1-dimensional foliations in the 3- dimensional space T 1 M. Together with the 1-dimensional foliation Z tangent to the geodesic flow, one now has 3 foliations E ss, E uu and Z that are independent and span the tangent spaces of T 1 M. Using clever ad hoc arguments that work only in dimension 2 Hopf showed that the foliations E ss, and E uu are C 1 in T 1 M if M has dimension 2 and

4 strictly negative Gauss curvature K. Hopf then devised a general method to prove ergodicity of the geodesic flow, which essentially has still not been improved upon. Moreover, Hopf realized that his method would generalize to arbitrary dimensions, provided that one could prove that the (n-1) - dimensional foliations E ss and E uu are still C 1 in T 1 M if M is a compact Riemannian manifold with strictly negative sectional curvature. Hopf was able to prove that these foliations are C 1 only in special cases such as constant negative sectional curvature, when these foliations are in fact C. The study of geodesic flows on compact negatively curved spaces languished until the mid or late 60 s when D. Anosov [Ano] overcame Hopf s technical problem with new methods and a new viewpoint. Putting the geodesic flow in the broader context of U- flows (later called Anosov), Anosov realized that one did not really need the foliations E ss and E uu to be C 1 to prove ergodicity of the flow, but only absolutely continuous. He succeeded in proving absolute continuity in the general Anosov setting, and he also showed that the foliations E ss and E uu are not always C 1. In fact, the foliations are rarely C 1, even in the special case of geodesic flows of compact negatively curved manifolds, so that Hopf s original method could not be generalized as planned. See sections VI and VII for further details. During the 1970 s Pesin [Pe 1-4] and others developed a theory of nonuniformly hyperbolic systems, and Pesin applied them to compact manifolds of nonpositive sectional curvature that today are called rank 1 manifolds (see section III G for a definition of rank). (In fact, Pesin considered an even more general class of manifolds.) In general, rank 1 manifolds have geodesic flows that in many ways are similar to those of manifolds of strictly negative sectional curvature, a special class of rank 1 manifolds. However, an arbitrary rank 1 manifold may contain a great deal of zero curvature, which introduces technical complications into proving ergodicity of the geodesic flow that have still not been completely overcome, even in dimension 2. The dynamical behavior of the geodesic flow in a compact, rank 1 manifold of nonpositive sectional curvature remains a major open problem. Spurred by the Mostow Rigidity Theorem [Mo], the attention of researchers in geometry and dynamical systems turned during the 80 s more toward rigidity phenomena such as the characterization of Riemannian symmetric spaces by various geometric and dynamical conditions. Some of these problems are described in section VII, along with many partial solutions that are outstanding even in the special cases considered. In section VIII we discuss the spectacular result of Besson - Courtois - Gallot [BCG 1, 2] that unified the earlier results and settled a number of the remaining open problems. For a more detailed description of this article see the table of contents. 4

5 5 II. Preliminaries on Riemannian manifolds A. Riemannian metric and Riemannian volume element. A C manifold X is a Riemannian manifold if for each point x of X there exists an inner product <, > x on the tangent space T x X and the assignment x <, > x is C ; that is, if V,W are C vector fields on X, then the function x <V(x), W(x)> is a C function on X. Define the norm v of a vector v in T x X in the usual way by v = < v, v > 1/2, and define the length of a C 1 curve c : [a,b] X by b L(c) = a c (t) dt. Define the distance d(x,y) between points x and y of X to be the infimum of the lengths of all curves joining x to y. The distance function d(x,y) is positive when x y and is called the Riemannian metric determined by the Riemannian structure { <, > x : x X}. If = ( 1,..., n ) : U (U) n is a local coordinate system defined on an open set U of X, then we obtain for each point x of U a positive definite symmetric matrix (g ij (x)) defined by g ij( x) = <, > (x). This is the classical description of x i x j the Riemannian structure <, > in local coordinates. A volume form for an n-dimensional, orientable C manifold X is a choice of a nowhere vanishing n - form ω on X, and the integral of a function f : X with compact support is defined to be the integral of fω over X. Now fix an orientation on an orientable Riemannian manifold X. If {e 1 (x),..., e n (x)} is an orthonormal basis for T x X that is consistent with the fixed orientation of X, and if {θ 1 (x),..., θ n (x)} is the corresponding dual basis of 1-forms, then the assignment x ω(x) = θ 1 (x)^... ^ θ n (x) is the Riemannian volume element determined by the Riemannian structure x <, > x. The definition of ω(x) depends only on the orientation of X and is independent of the oriented orthonormal basis {e 1 (x),..., e n (x)} of T x X. B. The Levi Civita connection and covariant differentiation along curves. Given a C vector field W on a Riemannian manifold X = {X, <, >} and a tangent vector v in T x X we wish to define the directional derivative of W in the direction v as a vector v W in T xx. If X = n with the usual inner product on each tangent space, and W is a vector field on n with component functions W 1,..., W n, then we

6 6 define v W to be the vector in T x n whose components are the standard directional derivatives D v W 1,..., D v W n. To define the directional derivative vw of a vector field W in an arbitrary Riemannian manifold X we follow the Koszul approach by stating as axioms certain properties that we wish to be satisfied and then showing that these axioms uniquely determine the definition of vw. All of these properties are clearly satisfied by vw as defined above in n with the standard inner products on each tangent space. We then derive the classical formulas for, the Levi Civita connection, in local coordinates using the Christoffel symbols. The Riemannian curvature tensor R can then be defined by, and it will be, but the classical Christoffel symbol formulation of R will be avoided. Axioms for the Levi Civita connection Let there be given C vector fields V, W on a Riemannian manifold X and arbitrary vectors v,w in T x X. Then there exists a unique vector v W in T xx such that the following properties are satisfied. 1) (Leibnizian property) If f : X is any C function, then v (fw) = v(f)w(x) + f(x) vw, where v(f) denotes the derivative of f at x in the direction v. 2) ( -linearity) a) v+w W = v W + w W av W = a ( v W) for any real number a b) v (V+W) = v V+ v W. v (aw) = a ( vw) for any real number a. 3) V W W V = [V,W] where [V,W] denotes the Lie bracket of V and W, and V W and W V denote the vector fields on X given by ( V W)(x) = V(x) W and ( W V)(x) = W(x) V. 4) v <V,W> = < v V, W(x) > + < V(x), v W >, where < V, W > is the C function on X defined by <V,W>(x) = < V(x), W(x) > and v <V,W> denotes the derivative of < V,W > in the direction v. Christoffel symbols and their formulas in local coordinates Let = ( 1,..., n ) : U (U) n be a local coordinate system defined on an open set U of X. For 1 i, j, k n we define C functions Γ k ij : U by

7 7 n / x / x i j = Σ Γij k / xk. k=1 The functions {Γ k ij} are called the Christoffel symbols determined by the Levi Civita connection. By 3) above and the fact that [ / x i, / x j ] = 0 it follows that the Christoffel symbols are symmetric in the two lower indices ; that is, Γ k ij = Γji k for all i, j, k. Using the axioms for above and the symmetry of the Christoffel symbols { Γ k ij} in the lower two indices it is routine to derive the classical formulas for the {Γij k } in terms of the functions g ij : U and their partial derivatives ; namely, Γ k ij = 1 2 s=1 Σ where g sk = (g 1 ) sk. n { gsj / x i + g is / x j g ij / x s ) gsk Covariant differentiation of vector fields along curves Let c : [a,b] X be a C curve in a Riemannian manifold X, and let Y : [a,b] TX be a C vector field along c; that is, Y(t) Tc(t) X for every t [a,b]. We wish to define a new vector field Y (t) along c in an axiomatic way analogous to the definition of the Levi Civita connection above. This notion is needed to define parallel translation of vectors along arbitrary curves. There exists for each C vector field Y(t) along c a unique C vector field Y (t) along c such that the following properties are satisfied. 1) (Leibnizian property) If f : [a,b] is any C function, then (fy) (t) = f (t) Y(t) + f(t) Y (t) for all t [a,b]. 2) (Y 1 + Y 2 ) (t) = Y 1 (t) + Y 2 (t) for any two vector fields Y 1, Y 2 along c. (ay) (t) = a Y (t) for all a and all t [a,b]. 3) If Y is the restriction to c of a vector field Y* on X, then Y (t) = c (t) Y* for all t [a,b]. 4) For any two vector fields Y 1, Y 2 along c the function t < Y 1 (t), Y 2 (t) > satisfies the equation d dt < Y 1 (t), Y 2 (t) > =< Y 1 (t), Y 2 (t) > + < Y 1 (t), Y 2 (t) > Remarks 1) An arbitrary curve c : [a,b] X may be highly singular or even constant so that not every vector field Y along c is the restriction to c of a vector field Y* of X. 2) For curves c : [a,b] U, where U is a coordinate neighborhood of X, one can define the covariant derivative Y (t) of a vector field Y(t) along c in terms of local

8 8 coordinate expressions involving the component functions of Y(t) and c(t) and the Christoffel functions restricted to c. We leave the derivation of such an expression to the diligent motivated reader. C. Parallel translation of vectors along curves Let c : [a,b] X be an arbitrary C curvein a Riemannian manifold X. A C vector field Y(t) along c is said to be parallel if the covariant derivative Y (t) is identically zero. Writing the equations for Y (t) 0 in local coordinates a standard differential equations argument shows that for every vector v in T c(a) X there exists a unique parallel vector field Y(t) along c such that Y(a) = v. The vector Y(b) is called the parallel translate of v along the curve c from c(a) to c(b). Note that if Y 1 (t) and Y 2 (t) are any two parallel vector fields along c, then the function t < Y 1 (t), Y 2 (t) > is constant in t by property 4) above. In particular the lengths of Y 1 (t) and Y 2 (t) and the angles between Y 1 (t) and Y 2 (t) are constant in t, justifying the name parallel translation. D. Curvature Riemannian curvature tensor Let X be a C Riemannian manifold. Given three C vector fields U,V and W on X we define a new vector field R(U,V)W = U ( V W) V ( U W) [U,V] W, where denotes the covariant derivative of the Levi Civita connection defined above. One can show that the value of R(U,V)W at a point x of X depends only on the values of U, V and W at x. Hence for each point x of X we obtain a trilinear map R : (T x X) x (T x X) x (T x X) T x X given by (u,v,w) R(u,v)w = {R(U, V) W}(x), where U, V and W are C vector fields with values u, v and w at x.. This is the Riemannian curvature tensor of X. We make no attempt to satisfy the reader who wishes to see the classical tensor expression in terms of Christoffel symbols and their derivatives. This expression would illustrate all too clearly how simple geometric concepts can be made to look both ugly and deep, on the one hand discouraging mathematicians from learning geometry and on the other hand impressing nonmathematicians unreasonably [V]. We will need the Riemannian curvature tensor to define the second order differential equation whose solutions are the Jacobi vector fields on geodesics γ. Jacobi

9 9 vector fields on a geodesic γ are basic objects that describe the first order behavior of the geodesics in a neighborhood of γ. It is a straightforward exercise to verify from the definitions that the curvature tensor R satisfies the following symmetry equations for all vectors u,v,w and z at a fixed point of X : R(u,v) w = R(v, u)w R(u, v) w + R(v, w) u + R(w, u) v = 0 < R(u, v) w, z > = < R(u, v) z, w > < R(u, v) w, z > = < R(w, z) u, v > Curvature operators Each vector v in T x X defines a curvature operator R v : T x X T x X by R v (u) = R(u, v) v. It follows from the symmetry equations above that each curvature operator R v is a symmetric linear transformation on T x X. Sectional curvature To each point x of a Riemannian manifold X and to each 2-dimensional subspace Π of the tangent space T x X one may assign a number K(Π) = < R(x, y) y, x > / x ^ y 2, where {x, y} is any basis of Π and x ^ y 2 = x 2 y 2 < x, y > 2. The number K(Π) is called the sectional curvature of Π, and it is independent of the basis {x, y} of Π. If X has constant sectional curvature c (i.e. K(Π) = c for all 2-planes Π), then the curvature tensor R has the following simple form : R(u, v) w = c{< v, w > u < u, w > v} for all vectors u, v and w in an arbitrary tangent space T x X. It is not difficult to show that the Euclidean space n of any dimension with the standard inner product on each tangent space has a curvature tensor that is identically zero, and hence n has zero sectional curvature. The sphere in n of radius a with the Euclidean inner product on each tangent space has constant sectional curvature 1 / a 2. The hyperbolic space of any dimension (see section III D below) has constant negative sectional curvature. A basic theorem in Riemannian geometry asserts that any simply connected Riemannian manifold X with constant sectional curvature c that is complete as a metric space must be isometric to one of the three examples above, depending upon the magnitude and sign of c. Sectional curvature may also be described in a simpler, if slightly imprecise, geometric fashion. Given a 2-plane Π in T x X let Σ Π denote the surface in X consisting

10 10 of the union of all geodesics in X that pass through x and are tangent to Π. Then K(Π) is the Gaussian curvature at x of the surface Σ Π. (Of course, this requires knowing what the Gaussian curvature is in an abtract 2-dimensional Riemannian manifold.) If one knows all sectional curvatures of 2-planes at a given point x of X, then, in principle, one can recover the Riemannian curvature tensor at x by a complicated polarization formula. Mean curvature of a hypersurface Let X be a Riemannian manifold of arbitrary dimension n, and let Y be a hypersurface in X, that is, a submanifold of codimension 1. Let U be one of the two C unit vector fields on Y that is orthogonal to Y, and define a linear transformation S : T y Y T y X by S(v) = v U for any vector v in T yy. It is not difficult to show that the image of S is contained in T y Y and that S : T y Y T y Y is a symmetric linear transformation. The trace of S is called the mean curvature of Y at y. E. Geodesics and geodesic flow A geodesic in a Riemannian manifold X is defined to be a C curve c : [a,b] X whose velocity vector field c (t) is a parallel vector field along c. The fact that c (t) is parallel implies immediately that a geodesic c(t) has constant speed since by the discussion above parallel vector fields have constant norm. This somewhat formal definition of geodesic also leads to a second order ordinary differential equation in local coordinates that involves the Christoffel symbols. The diligent reader may derive this equation as an exercise, and the less diligent reader may simply believe the statement above or look it up somewhere. The point is that the existence and uniqueness theory of ordinary differential equations shows that for every point x in X and every vector v in T x X there exists a positive number ε = ε(x,v) and a unique geodesic γ v : ( ε, ε) X such that γ v (0) = x and γ v (0) = v. Geodesic completeness and the theorem of Hopf - Rinow In general a geodesic c(t) may not be defined on the whole real line. For example, remove the origin from the Euclidean plane with its standard inner product <, > on each tangent space. The geodesics of the new space X = 2 origin still are straight lines, but those that used to run through the origin can no longer do so and are therefore not

11 11 defined on all of. In most cases Riemannian geometers only consider Riemannian manifolds that are geodesically complete; that is, every geodesic is defined on all of. This is guaranteed if X is compact. In general the following theorem of Hopf and Rinow characterizes geodesic completeness. See for example [Mi] for a proof. Theorem Let X be a Riemannian manifold. Then the following conditions are equivalent : 1) All geodesics of X are defined on. 2) For some point x of X all geodesics of X that pass through x are defined on. 3) If d denotes the Riemannian metric of X, then the metric space (X, d) is complete; that is, every Cauchy sequence converges. 4) Every closed, bounded subset of X is compact. Geodesics as critical points of arc length Next we give a geometric description of geodesics that conforms better to the usual notion of geodesics as shortest curves. Note that the parametrization of a geodesic is important in Riemannian geometry. Geodesics must always have constant speed and to be a geodesic it is not enough for a curve c to be the shortest connection between its endpoints. (However, any constant speed reparametrization of c will then be a geodesic by the discussion that follows.) Given any two points p, q in a Riemannian manifold X let Ω pq denote the collection of all piecewise C curves c : [0,1] X, and let L : Ωpq denote the length function. The geodesics from p to q can be described as the critical points of the length functional L on Ω pq in a sense we now make precise. A fixed endpoint variation of a piecewise C curve c : [0,1] X is a C family of piecewise C curves ct : [0,1] X, ε < t < ε, such that c o = c and c t (0) = c(0), c t (1) = c(1) for all t. (Formally, we require the map F : [0,1] x ( ε, ε) X given by F(s,t) = c t (s) to be C.) The curve c is said to be a critical point of the variation {ct } if L (0) = 0, where L(t) = L(c t ), the length of c t, for all t. If c : [0,1] X has constant speed, necessary if c is to be a geodesic, then c is a geodesic of X if and only if c is a critical point of every variation {c t } of c. See for example [Mi] for a proof. In particular, if c has constant speed and is a shortest curve in Ω pq, then c is a geodesic from p to q. However, not every geodesic from p to q is a shortest curve from p to q. For example, on the standard 2-sphere of radius 1 where all great circles have length 2π any

12 12 great circle is a shortest curve until the antipode of the starting point is reached, but beyond that the great circle is no longer a shortest curve although it is still a geodesic. One may think of Ω pq as an infinite dimensional manifold in which the points of Ω pq are curves from p to q, parametrized on [0,1]. The tangent space T c Ω pq at the curve c consists of all initial velocities of piecewise C fixed endpoint variations of c ; that is, T c Ω pq is the infinite dimensional real vector space of piecewise C vector fields Y(t) on c such that Y(0) = Y(1) = 0. The nature of a geodesic c as a critical point of the arc length functional L can be investigated by studying the second derivative L (0) for all possible fixed endpoint variations of c. This second derivative information is encoded in a symmetric bilinear form I : T c Ω pq x T c Ω pq called the index form. The form I is positive definite if and only if c is shorter than all nearby curves of Ω pq. If I(Y,Y) is negative for some vector field Y in T c Ω pq, then one may use Y to construct a fixed endpoint variation {c t } of c in which all nearby curves c t are shorter than c; i.e. L(c t ) < L(c) if t is small. It turns out that the maximum dimension of a subspace W of T c Ω pq on which I is negative definite is always finite, and the dimension of W can be described precisely by the Morse Index Theorem in terms of conjugate points of c, which we define below. See for example [Mi] for details, including a definition and properties of the index form I. Geodesic flow Let X be a geodesically complete Riemannian manifold; that is, all geodesics of X are defined on. Let T 1 X denote the collection of unit vectors tangent to X. We have seen that for every vector v in T 1 X there exists a unique geodesic γ v in X with initial velocity v. For each number t in we define a diffeomorphism g t : T 1 X T 1 X by g t (v) = γ v (t), the velocity of γ v at time t. Since X is geodesically complete the diffeomorphisms are defined for all t in, and the collection of diffeomorphisms {g t } is a 1-parameter group; that is, g t ο g s = g s ο g t = g t+s for all s, t in. The diffeomorphisms {g t } are called the geodesic flow in T 1 X. F. Riemannian exponential map and Jacobi vector fields Let X be a geodesically complete Riemannian manifold. For each point x of X there exists an important map exp x : T x X X called the Riemannian exponential map. Given a vector v in T x X, not necessarily of length 1, we define exp x (v) = γ v (1), the value of γ v at time 1. It is not difficult to show that exp x (t v) = γ v (t) for all t in so that exp x

13 13 maps the straight lines through the origin in T x X into the geodesics of X that start at x. We shall see shortly that the critical points of exp x have an important geometric meaning. Let c : [a,b] X be a geodesic of X, which for convenience we assume to have unit speed. A C 2 vector field Y(t) on c is defined to be a Jacobi vector field if it satisfies the second order linear differential equation (J) Y (t) + R (Y(t), c (t)) c (t) = 0 on [a,b]. Here Y (t) denotes the covariant derivative of Y(t) along c, and Y (t) denotes the covariant derivative of Y (t) along c, or equivalently, the second covariant derivative of Y(t) along c. R denotes the curvature tensor of X defined above. If we define R(t) to be the symmetric curvature operator v R(v, c (t)) c (t) determined by c (t), then we may rewrite the Jacobi equation (J) above as Y (t) + R(t) (Y(t)) = 0 By using an orthonormal family {E i (t)} of parallel vector fields along c(t) the equation above may be converted into a second order matrix differential equation where Y(t) is an n x 1 column vector and R(t) is a symmetric n x n matrix. The standard theory of ordinary differential equations says that given a vector v at a point x of X and any two vectors w,w* in T x X there exists a unique Jacobi vector field Y(t) along the geodesic γ v such that Y(0) = w and Y (0) = w*, where as usual Y (t) denotes the covariant derivative of Y along γ v. If we let J(γ v ) denote the collection of all Jacobi vector fields on γ v, then it is apparent from the linear equation (J) and the remarks just made that J(γ v ) is a vector space of dimension 2n, where n is the dimension of X. Jacobi vector fields as geodesic variation vector fields It is useful to describe an equivalent geometric definition of Jacobi vector fields. An (arbitrary) C variation of a C curve c : [a,b] X is a C family of curves {c t }, ε < t < ε, such that c o = c. (We no longer require that the curves {c t } have the same endpoints as in the discussion of geodesics above.) A C variation of c determines a variation vector field Y(s) on c ; define Y(s) to be the initial velocity of the curve t c t (s) for each fixed s in [a,b]. If c is a geodesic of X, then a C 2 vector field Y(t) on c is a Jacobi vector field if and only if Y is the variation vector field of a geodesic variation of c; that is, a C variation {ct } of c in which each of the curves c t is a geodesic of X. Conjugate points and critical points of the exponential map

14 14 Let c : [a,b] X be a nonconstant geodesic in a (geodesically) complete Riemannian manifold. The points c(a) and c(b) are conjugate along c if there exists a nonzero Jacobi vector field Y(t) along c such that Y(a) = 0 and Y(b) = 0. It is not difficult to show that Y(t) can be represented as the variation vector field of a geodesic variation {c t } such that c t (a) = c(a) for all t. The fact that Y(b) = 0 means that the curve σ(t) = c t (b) has initial velocity zero and may be regarded as constant to first order. Geometrically one may think of the geodesics {c t } as a family of light rays emerging from the point c(a) and focusing at the point c(b). This behavior is typical of spaces with positive sectional curvature if one thinks of great circles on the sphere emerging from the north pole and meeting again at the south pole. The existence of conjugate points along a geodesic is related to the critical points of the exponential map as follows. Let x be a point of a complete Riemannian manifold X, and let v be a vector in T x X. Then v is a critical point of the Riemannian exponential map exp x : T x X X if and only if the points γ v (0) = x and γ v (1) are conjugate along γ v. To see this, let ξ be a nonzero vector in T x X such that dexp x (ξ v ) = 0, where ξ v denotes the initial velocity of the curve t v + t ξ in T x X. If c t is the geodesic starting at x with initial velocity v + t ξ and if Y is the variation vector field of the geodesic variation {c t }, then Y(t) is a nonzero Jacobi vector field such that Y(0) = 0 and Y(1) = 0. If X is a complete Riemannian manifold with nonpositive sectional curvature (i.e. K 0), then we shall see that any nonzero Jacobi vector field Y(t) with Y(0) = 0 on a geodesic c(t) grows in norm at least as fast as a linear function. It follows that no nonzero Jacobi vector field can vanish twice on a geodesic, and by the remarks above this means that every exponential map exp x : T x X X is nonsingular and, in fact, a covering map. In the case that X is simply connected it follows that each exponential map exp x : T x X X is a diffeomorphism, which means geometrically that for any two distinct points x and y in X there is a unique geodesic (up to constant speed reparametrizations) joining x to y. This is the theorem of Cartan - Hadamard. G. Isometries and local isometries Let X and Y be Riemannian manifolds. An isometry p : X Y is a diffeomorphism such that < dp x (v), dp x (w) > = < v, w > for all vectors v,w in T x X and all points x of X. Clearly, p 1 : Y X is also an isometry. It is not difficult to show that an isometry p carries geodesics to geodesics and Jacobi vector fields on a geodesic c(t) to Jacobi vector fields on the geodesic (p ο c)(t). If Π is a 2-dimensional subspace of a tangent space T x X, then the sectional curvature of Π equals that of dp(π) T p(x) Y.

15 15 In general, isometries preserve all possible geometric properties defined by the Riemannian structure <, >. From the definition of the Riemannian metric d it follows that if p : X Y is an isometry, then d(p(x), p(x*)) = d(x, x*) for all points x,x* in X. Conversely, if p : X Y is a diffeomorphism such that d(p(x), p(x*)) = d(x, x*) for all points x,x* in X, then it is not difficult to show that p is an isometry. In summary, one may describe the isometries as the distance preserving diffeomorphisms. A C map p : X Y is called a local isometry if < dpx (v), dp x (w) > = < v, w > for all vectors v,w in T x X and all points x of X (i.e. we drop the condition that p be a diffeomorphism). Any local isometry is nonsingular, and hence by the inverse function theorem for any point x of X there exist open neighborhoods U of x in X and V of p(x) in Y such that p : U V is a diffeomorphism. The map p : U V is therefore an isometry between the Riemannian manifolds U and V, which explains the term local isometry. All of the geometric quantities discussed above (geodesics, sectional curvature, Jacobi vector fields etc.) are defined by equations that depend on local information in the Riemannian manifold. Hence local isometries also preserve all locally defined geometric quantities such as geodesics, sectional curvature and Jacobi vector fields. Moreover, the preimages of geodesics in Y under a local isometry p : X Y are geodesics in X. H. Geometry of the tangent bundle with the Sasaki metric Let X be a C differentiable manifold and let TX denote the tangent bundle of X, which consists of the union of all tangent spaces T x X, for x X. There is a natural projection map π : TX X that sends a vector v in T x X to its basepoint x in X. The fibers of π are the tangent spaces T x X as x ranges over X, and hence TX is a manifold of dimension 2n, where n is the dimension of X. If X is in addition a Riemannian manifold, then the Riemannian metric on X induces a Riemannian metric on TX that is called the Sasaki metric. To describe it we first define the connection map : TX X, which unlike the projection map π : TX X depends on the Riemannian structure of X. For further properties of the connection map see [E5] and [GKM]. Given a vector ξ in T v (TX) let v(t) be a curve in TX such that v(0) = v and v(t) has initial velocity ξ. Let c(t) = π(v(t)) so that v(t) may be regarded as a C vector field on the curve c(t) in X. Now define (ξ) to be the vector v (0) in T x X, where x = π(v), the base point of v, and v (t) denotes the covariant derivative of v(t) along c(t). An

16 16 explicit computation of (ξ) in local coordinates shows that the definition of (ξ) does not depend on the curve v(t) chosen but only its initial velocity. Following our earlier practice, we leave the definition of (ξ) in local coordinates as an exercise for the diligent reader. The subbundles Ker ( ) and Ker (dπ) of TX are called the horizontal and vertical subbundles respectively. It is not hard to see that TX is the direct sum of the horizontal and vertical subbundles. We are now ready to define the Sasaki metric in TX. Given vectors ξ, η in T v (TX) we define < ξ, η > TX = < dπ(ξ), dπ(η) > X + < (ξ), (η) > X, where π : TX X and : TX X denote the projection and connection maps respectively. This bilinear form on TX is positive definite since Ker ( ) and Ker (dπ) are disjoint except for {0}. Identification of T v (TX) with J(γ v ) For each vector ξ in T v (TX) we let Y ξ be the unique Jacobi vector field on γ v such that Y ξ (0) = dπ(ξ) and Y ξ (0) = (ξ). It is not difficult to show that the map ξ Y ξ is an isomorphism of T v (TX) onto J(γ v ), the space of Jacobi vector fields on γ v. Geometrically, one may also describe this situation as follows. Given a vector ξ T v (TX) let v(t), ε < t < ε, be a C curve in TX such that v(0) = v and v(t) has initial velocity ξ. If c t denotes the geodesic with initial velocity v(t), then {c t } is a geodesic variation of the geodesic c o = γ v, and the corresponding variation vector field is a Jacobi vector field on γ v that is easily seen to be the Jacobi vector field Y ξ (t) defined analytically above. If T 1 X denotes the bundle of unit tangent vectors of X, a codimension 1 submanifold of TX, then the isomorphism ξ Y ξ carries T v (T 1 X) onto J*(γ v ), the codimension 1 subspace of J (γ v ) consisting of those Jacobi vector fields Y(t) on γ v such that Y (t) is orthogonal to γ v (t) for all t. Note that each map g t : TX TX leaves invariant T 1 X since g t (v) = γ v (t) for any unit vector v and the velocity vector field γ v (t) has constant norm since it is a parallel vector field on γ v. Now let {g t } denote the geodesic flow on TX or T 1 X. If ξ is any vector in T v (TX), with corresponding Jacobi vector field Y ξ on γ v, then it follows from the definitions that dg t (ξ) 2 = Y ξ (t) 2 + Y ξ (t) 2 for all t. This fact allows one to estimate the growth rate of dg t (ξ) by using the Jacobi equation to estimate the growth rate of Y ξ (t), which, when the sectional curvature of X is strictly negative, is essentially the same as that of Y ξ (t) for those Jacobi vector fields Y ξ most important

17 17 for the geodesic flow. When the sectional curvature is bounded above by a negative constant a 2, then comparisons to Jacobi vector fields in a space of constant sectional curvature a 2 yield sharp estimates for the growth rate of Y ξ. See III B for further details. We now combine the discussion of subsections F and G to obtain the following useful result that relates Jacobi vector fields to the differentials of the exponential maps. Proposition Let X be a complete Riemannian manifold, and let exp x : T x X X be the exponential map at a point x of X. Let v, ξ be any vectors in T x X and let ξ v T v (T x X) T v (TX) denote the initial velocity of the curve t v(t) = v + t ξ. Then dexp x (ξ v ) = Y(1), where Y(t) is the unique Jacobi vector field on the geodesic c(t) = exp x (tv) such that Y(0) = 0 and Y (0) = ξ. Proof For t < ε and s define c t (s) = exp x (s (v + t ξ). Each curve c t is a geodesic, with initial velocity v(t) = v + t ξ and the variation vector field is the Jacobi vector field Y(s) on c(s) = c o (s) = exp x (sv) such that Y(0) = dπ (ξ) and Y (0) = (ξ), where : TX X denotes the connection map. However, dπ (ξ) = 0 since dπ (ξ) is the initial velocity of the constant curve π(v(t)) = {x}. By definition (ξ) = Z (0), the covariant derivative at t = 0 of the vector field Z(t) = v + t ξ on the constant curve π(v(t)) = {x}, but this is precisely ξ. The invariant volume element on TX and T 1 X There is a natural volume element on the tangent bundle TX of a Riemannian manifold X and a closely related natural volume element on the unit tangent bundle T 1 X. In both cases these are the Riemannian volume elements as determined by the Sasaki metric defined above, but they also have other and better known descriptions. We explain briefly. Given a differentiable manifold X there is a canonical 1-form θ on the cotangent bundle T*X consisting of the union of all cotangent spaces {T*X x = Hom (T x X, ), x X} together with the natural projection π* : T*X X that sends an element of T x *X to x. For ω T x *X and ξ T ω (T*X) we define θ(ξ) = ω(dπ*(ξ)). If Ω = dθ, then Ω is a symplectic 2-form on T*X and in particular Ω... Ω (n times) is a nowhere vanishing volume element on T*X, where n = dim X. If X has a Riemannian structure <, >, then there is a natural diffeomorphism

18 18 : TX T*X : given a vector v in T x X let (v) = ω v be that element in T x *X such that ω v (ξ) = < ξ, v > for all vectors ξ in T x X. Under this identification of TX and T*X we let θ and Ω also denote the corresponding 1 and 2-forms in TX as well as their restrictions to the unit tangent bundle T 1 X. One can show that the geodesic flow {g t } in either TX or T 1 X leaves invariant the forms θ and Ω ; that is, (g t )*θ = θ and (g t )*Ω = Ω for all t. The Riemannian volume elements in TX and T 1 X with respect to the Sasaki metric may now be described respectively as Ω... Ω (n times) and θ (Ω... Ω) (n-1 times). III. Manifolds of nonpositive sectional curvature Notation : In the sequel we will use M to denote a complete Riemannian manifold (usually compact) and M ~ to denote its universal Riemannian cover; that is, M ~ is the unique simply connected differentiable manifold that covers M, and M ~ is equipped with the unique inner products on tangent spaces that make the differential maps of the covering map p : M ~ M linear isometries. A. Definition of nonpositive curvature by triangle comparisons Let be a triangle in a complete Riemannian manifold X whose sides are geodesics and also shortest curves connecting the vertices A, B and C. Let the sides of have lengths a, b and c as shown in the figure below. Now let * be a triangle in the Euclidean plane with the same side lengths a, b and c and opposite vertices A*, B* and C*. Let p and p* be points in and * that belong to corresponding sides and have the same distances from the opposite vertices, as shown in the figure below. Using Jacobi vector field arguments one can prove that if A and A* are the vertices opposite p and p*, then d(a, p) d(a*, p*) for all triangles and * if and only the sectional curvature of X is nonpositive. Similarly, X has sectional curvature c 2 < 0 if and only if analogous triangle comparisons with a hyperbolic space of constant sectional curvature c 2 < 0 are valid. Conversely, one can use the triangle comparison definition above to define metric spaces X of curvature 0 or c 2 < 0, often when no differentiable structure for X exists, provided that the metric spaces X admit geodesics with reasonably nice properties. This generalization of Riemannian geometry to (possibly) nondifferentiable spaces of nonpositive or strictly negative curvature has been extremely fruitful for geometers, but it

19 19 lies at some distance from smooth dynamical systems and will not be treated in these notes. B. Growth of Jacobi vector fields Let Y(t) be a Jacobi vector field on a unit speed geodesic c(t) in a complete Riemannian manifold X. If Y tan (t) and Y (t) denote the components of Y(t) that are tangent to and orthogonal to c (t), respectively, then it follows from the Jacobi equation that Y tan (t) and Y (t) are also Jacobi vector fields on c(t). If Y(t) = f(t) c (t) is a tangential Jacobi vector field on c(t), then it follows easily from the Jacobi equation that f(t) is a linear function at + b for suitable constants a and b. Therefore, in estimating the growth of the norm of a Jacobi vector field, the only interesting case occurs when Y(t) is orthogonal to c (t). Here the influence of the sign of the sectional curvature is significant. Let Y(t) be a Jacobi vector field on a unit speed geodesic c(t) in a complete Riemannian manifold M with sectional curvature K 0. If f(t) = Y(t) 2 = <Y(t), Y(t) >, then a direct computation shows that f (t) 0 for all t; that is, f is a convex function on. Specifically, we compute f (t) = 2<Y(t), Y (t) > and by using the Jacobi equation we obtain f (t) = 2{ Y (t) 2 + <Y (t),y(t) >} = 2{ Y (t) 2 < R(Y(t), c (t)) c (t),y(t) >} = 2{ Y (t) 2 K(Y(t), c (t)) Y(t) ^ c (t) 2 } 0, where Y (t) and Y (t) denote the first and second covariant derivatives of Y(t) along c(t) and K(Y(t), c (t)) denotes the sectional curvature of the 2-plane spanned by Y(t) and c (t). (If Y(t) and c (t) are collinear then we adopt the convention that K(Y(t), c (t)) = 0 since in this case R(Y(t), c (t)) c (t) = 0 by symmetries of the curvature tensor described above.) From the properties of convex functions from to and the arguments of the previous paragraph we may immediately conclude the following. Proposition Let M be a complete Riemannian manifold with sectional curvature K 0, and let Y(t) be a Jacobi vector field on a unit speed geodesic c(t) of M. Then either 1) Y(t) is nondecreasing on, and Y(t) + as t, or 2) Y(t) is nonincreasing on, and Y(t) + as t, or 3) Y(t) is constant on and Y(t) is a parallel vector field on c(t). The proposition above is a simple example of the important role that convex functions play in the geometry of complete manifolds M with sectional curvature K 0. A similar but more difficult computation for the function g(t) = Y(t), where Y(t) is a

20 20 Jacobi vector field on a unit speed geodesic c(t), shows that g (t) 0 whenever g(t) 0. If Y(0) = 0, then it is not difficult to show that g (0 + ) = Y (0). We obtain (*) If Y(t) is a Jacobi vector field on a unit speed geodesic c(t) with Y(0) = 0, then Y(t) t Y (0) for all t > 0. In particular, Jacobi vector fields Y(t) that start at zero in a manifold M of nonpositive sectional curvature grow in norm at least as fast as Euclidean Jacobi vector fields with the same initial rate of increase. The inequality (*) is the key fact needed to prove the triangle comparison result stated above, but we shall not pursue this. Growth rates in manifolds with K c 2 < 0 Generally speaking, if one can solve equations like the Jacobi equation in spaces where the sectional curvature is identically c 2 < 0, then one can obtain estimates for the solutions of the corresponding equations in manifolds with K c 2 < 0 or K c 2. In the case of the Jacobi equation, the mechanism for obtaining such estimates is the Rauch Comparison Theorem (cf. [CE]), which is a generalization of the classical Sturm comparison theorem that applies to the growth of solutions to the scalar equation y (t) + k(t) y(t), where k(t) c for all t or k(t) c for all t and c is some real number. To illustrate with a specific example, let Y(t) be a nonzero Jacobi vector field orthogonal to c (t) on a unit speed geodesic c(t) in a Riemannian manifold M with constant sectional curvature K λ. The norm y(t) = Y(t) of a Jacobi vector field satisfies the scalar equation y (t) + λ y(t) = 0, for which the solutions are easy to find. If K c 2 and if Y(0) = 0, then Y(t) = sinh(ct) Y (0) for all t 0, where sinh (t) = (1/2) (e t e t ) is the hyperbolic sine. In general the Rauch Comparison Theorem yields the following : Let M be a complete Riemannian manifold with sectional curvature satisfying b 2 K a 2, where a and b are positive constants. If Y(t) is a nonzero Jacobi vector field orthogonal to c (t) on a unit speed geodesic c(t) of M with Y(0) = 0, then (*) (1/a) sinh (at) Y (0) Y(t) (1/b) sinh(bt) Y (0) for all t 0. In other words, Y(t) increases at least as fast as in the constant curvature case corresponding to the upper curvature bound and no faster than the constant curvature case corresponding to the lower curvature bound. The discussion above of the geodesic flow states that dg t (ξ) 2 = Y ξ (t) 2 + Y ξ (t) 2 for all t, where Y ξ is the Jacobi vector field with Y ξ (0) = dπ (ξ) and Y ξ (0) = (ξ) and is the connection mapping. In most important geometric situations Y ξ (t) is at most a bounded multiple of Y ξ (t) for sufficiently large t so

21 21 that dg t (ξ) is at most a bounded multiple of Y ξ (t) for sufficiently large t. For example, if M has constant sectional curvature K c 2, then y(t) = Y ξ (t) satisfies the scalar equation y (t) c 2 y(t) = 0. If Y ξ (0) = 0, then Y ξ (t) = y (t) = c {cosh (ct) / sinh (ct)} y(t) 2c y(t) = 2c Y ξ (t) if t is sufficiently large. If the sectional curvature of M satisfies b 2 K c 2, then the Rauch Comparison Theorem yields c {cosh (ct) / sinh (ct)} Y ξ (t) Y ξ (t) b {cosh (bt) / sinh (bt)} Y ξ (t) for all t. See also the lemma at the end of VI A. C. The Riemannian exponential map is a covering map. The theorem of Cartan Hadamard. Let M be a complete Riemannian manifold with sectional curvature K 0. Let m be any point of M, and let exp m : T m M M denote the Riemannian exponential map at m. We saw earlier at the end of II F that if v, ξ are any tangent vectors in T m M and if ξ v T v (T m M) denotes the initial velocity of the curve t v + t ξ, then dexp m (ξ v ) = Y(1 ), where Y(t) is the Jacobi vector field on γ v (t) = exp m (tv) such that Y(0) = 0 and Y (0) = ξ. By the discussion above in III B we saw that Y(t) t Y (0) = t ξ for all t 0. Evaluating at t = 1 we obtain (*) dexp m (ξ v ) ξ for all ξ v Τ v (T m M) or equivalently : If γ(t) is any C 1 curve in T m M, then L(γ) L(exp m ο γ). In particular, exp m : T m M M is a nonsingular map. In fact, we can say even more : Proposition Let M be a complete Riemannian manifold with sectional curvature K 0. Let m be any point of M, and let exp m : T m M M denote the Riemannian exponential map at m. Then exp m is a covering map. If M is simply connected, then exp m is a diffeo - morphism of M onto n, n = dim M. Corollary (Theorem of Hadamard - Cartan) Let M be a complete Riemannian manifold with sectional curvature K 0. For any two distinct points p and q there exists a unique (up to parametrization) geodesic γ pq joining p to q. Proof of the Corollary Given distinct points p and q of M it suffices to show that there exists a unique geodesic γ : [0,1] M such that γ(0) = p and γ(1) = q. By the definition and discussion

22 22 of the exponential map, every geodesic γ(t) with γ(0) = p has the form γ(t) = exp p (tv) for some tangent vector v of T p M. Let γ(t) and γ*(t) be two geodesics parametrized on [0,1] that join p to q, and choose vectors v and v* in T p M such that γ(t) = exp p (tv) and γ*(t) = exp p (tv*) for all t. Then q = γ(1) = exp p (v), and the same argument shows that q = exp p (v*). Hence v = v* and γ = γ* since exp p : T pm M is a diffeomorphism. Proof of the Proposition We shall need the following result from Riemannian geometry whose proof we omit. Lemma Let X and Y be complete Riemannian manifolds, and let p : X Y be a local isometry ; that is, < dp x (v), dp x (w) > = < v, w > for all vectors v and w in T x X and any point x of X. Then p is a covering map. We now prove the Proposition above. Fix a point m of M. By the inequality (*) above the exponential map exp m : T m M M is nonsingular. (In particular, exp m is a local diffeomorphism by the inverse function, which is a necessary condition for a C covering map.). Since exp m : T m M M is nonsingular, we may give the Euclidean space T m M the unique Riemannian structure <, > such that < dexp m (v), dexp m (w) > = < v, w > for all vectors v and w in T m M; i.e. the unique Riemannian structure <, > that makes exp m : T m M M a local isometry. See II G. By hypothesis M is a complete Riemannian manifold, and hence by the lemma above it suffices to show that T m M with the Riemannian metric <, > is a complete Riemannian manifold. By properties of the exponential map (cf. the discussion in II G) the preimages of the geodesics in M that start at m are precisely the straight lines in T m M that pass through the origin in T m M; that is the curves t t v for an arbitrary vector v in T m M. It is a well known fact in Riemannian geometry that local isometries map geodesics to geodesics and that the preimages of geodesics in the range manifold are geodesics in the domain manifold. In the present case it means that the straight lines through the origin in T m M are geodesics with respect to the Riemannian structure <, > defined on T m M. Since these straight line geodesics of <, > are defined on it follows by one of the equivalent formulations of the Hopf - Rinow theorem (see II E above) that T m M with the Riemannian structure <, > is complete. By the lemma above this completes the proof that exp m : T m M is a covering map for every point m of M. Finally, any C covering map between simply connected spaces must be a diffeomorphism, which completes the proof of the Proposition.