The Geodesic Flow

Introduction

The geodesic flow of a Riemannian manifold $(M,g)$ is the flow on the unit tangent bundle $SM$ that carries a unit vector to the velocity of the geodesic it generates: $\varphi_t(v)=\dot\gamma_v(t)$, where $\gamma_v$ is the geodesic with $\dot\gamma_v(0)=v$ and $t\mapsto\gamma_v(t)$ is parametrised by arclength. It is the flow of the geodesic spray, equivalently the Hamiltonian flow of the kinetic energy on the cotangent bundle, and it is the single object in which Riemannian geometry, the Hamiltonian formalism of the calculus of variations and smooth ergodic theory meet: the curvature of $g$ encodes itself in the derivative of the flow through the Jacobi equation, the volume form of the Sasaki metric gives the flow an invariant measure, and the question of how the geodesics distribute themselves is precisely the question of the ergodicity of that measure. If the curvature is negative, the answer is complete: the flow is Anosov, hence topologically mixing and structurally stable, it is ergodic and mixing for the Liouville measure by the argument of Hopf, the entropy of that measure is the sum of the positive Lyapunov exponents by Pesin's formula, the topological entropy equals the volume entropy by the theorem of Dinaburg and Manning, and the two agree for the space forms of constant curvature, its closed geodesics obey a prime geodesic theorem, and its periodic orbits equidistribute. If the curvature vanishes, as on a flat torus, the flow is completely integrable and the ergodicity fails on the whole bundle; between the two the nonpositive curvature brings the theory of Pesin, of the nonuniform hyperbolicity and of the invariant manifolds.

The article begins with the definition of the geodesic flow, the geodesic spray, the invariance of the Liouville measure and the identification with the Hamiltonian flow of the kinetic energy; the examples are the flat torus, the round sphere, the surfaces of revolution and the hyperbolic surfaces. The differential geometry of the flow follows: the Jacobi equation, the conjugate points, the behaviour of the Jacobi fields under negative curvature and the hyperbolicity it produces, with the stable and unstable distributions of the flow described by the horospheres. The Liouville measure and the ergodic theorems are then developed: the invariant measure, the Hopf argument for the ergodicity of the geodesic flow of a compact negatively curved manifold, the mixing, the entropy and its identification with the volume entropy by Dinaburg and Manning, and the Pesin formula for the entropy in terms of the Lyapunov exponents. The periodic orbits follow: the closed geodesics, the length spectrum, the prime geodesic theorem, the Ruelle and Selberg zeta functions, and the equidistribution of the closed geodesics. The article closes with the homogeneous dynamics of the locally symmetric spaces of negative curvature, where the flow becomes a homogeneous flow on $\Gamma\backslash G$ and Ratner's theorems apply, with the rigidity theorems of Mostow and the entropy rigidity, and with the flat and integrable examples and the open problems of the nonpositive curvature.

The geodesics, the curvature, the Jacobi fields, the conjugate points and the exponential map are those of Curvature and Geodesics and Riemannian Geometry, developed there and used here; the hyperbolic plane, its isometries and the hyperbolic surfaces are those of Hyperbolic Geometry; the smooth manifolds, the vector fields, the flows and the tangent bundle are those of Smooth Manifolds and Differential Geometry. The geodesic flow as a Hamiltonian system is that of Lagrangian and Hamiltonian Systems and The Calculus of Variations; the hyperbolicity, the Lyapunov exponents, the Pesin formula, the SRB measures and the Markov partitions are those of Hyperbolic Dynamics and Anosov Systems, and the flows, the Poincaré maps and the structural stability are those of Smooth Dynamical Systems; the recurrence, the entropy and the mixing are those of Topological Dynamics. The ergodicity, the Birkhoff theorem and the ergodic decomposition are those of Ergodic Theory; the homogeneous flows, the unipotent dynamics and the equidistribution are those of Homogeneous Dynamics, Ratner's Theorems and Equidistribution; the lattices and the locally symmetric spaces are those of Lattices in Lie Groups. The continued fraction coding of the modular flow is that of Dynamics and Number Theory.

No physics is invoked; the Hamiltonian formalism is used as a mathematical structure and no physical system is named.

The Geodesic Flow

The Flow of the Geodesic Spray

Definition. Let $(M,g)$ be a Riemannian manifold with Levi-Civita connection $\nabla$, and let $\pi:TM\to M$ be its tangent bundle. The geodesic spray is the vector field $G$ on $TM$ whose integral curves are the velocity fields of the geodesics, $G(v)=\frac{D}{dt}\big|_{t=0}\dot\gamma_v(t) \in T_vTM$; in local coordinates $(x,\xi)$ of $TM$ the spray is $\dot x^k=\xi^k$, $\dot\xi^k=-\Gamma^k_{ij}(x)\xi^i\xi^j$ and the geodesics are the projections of its integral curves. The geodesic flow is the restriction to the unit tangent bundle

$$ SM=\{v \in TM:\|v\|_g=1\} $$

of the flow of $G$: $\varphi_t(v)=\dot\gamma_v(t)$, where $\gamma_v$ is the geodesic with $\gamma_v(0)=\pi(v)$ and $\dot\gamma_v(0)=v$, parametrised by arclength. When $M$ is complete the flow is defined for all $t \in\mathbb{R}$ and $SM$ is a compact manifold if $M$ is compact.

Theorem (Hamiltonian form and invariant measure). The geodesic flow is the Hamiltonian flow of the function $H(q,p)=\frac12\|p\|_g^2$ on the cotangent bundle $T^*M$ with the canonical symplectic form, restricted to the level set $H=\frac12$; equivalently it is the Euler–Lagrange flow of the kinetic energy $\frac12\int\|\dot\gamma\|^2dt$ on the space of the paths. The Liouville measure on $SM$, the Riemannian volume of the Sasaki metric, is invariant under $\varphi_t$, and the flow preserves the contact form $\alpha=\langle p,dq\rangle$ on the level set, so that the flow is the Reeb flow of a contact structure and the Liouville measure is its volume.

Proof. The Hamiltonian of the kinetic energy in the metric is $H(q,p)=\frac12g^{ij}(q)p_ip_j$, and Hamilton's equations are $\dot q^k=\partial H/\partial p_k=g^{kj}p_j$ and $\dot p_k=-\partial H/\partial q^k=-\frac12\partial_kg^{ij}p_ip_j$; eliminating $p=g(\dot q,\cdot)$ gives the geodesic equation, so the two flows agree. The invariance of the Liouville measure is the Hamiltonian volume preservation of Lagrangian and Hamiltonian Systems; the contact form restricts from the symplectic potential and its Reeb field is the spray after the Legendre transform.

Example (the flat torus). On $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$ with the flat metric the geodesics are the straight lines, and the geodesic flow is $\varphi_t(x,v)=(x+tv,v)$ with $v$ a unit vector; the flow is completely integrable, its orbits are contained in the invariant tori of constant $v$, and it is ergodic on such a torus exactly when the direction $v$ generates a dense subgroup — the Kronecker flow of Topological Dynamics — but not on the whole of $S\mathbb{T}^n$, on which it is not ergodic.

Example (the round sphere). On $S^n$ with the round metric the geodesics are the great circles, every orbit of the flow is closed of length $2\pi$, and the flow is periodic; its invariant measures are the rotation-invariant measures on $SS^n$ and the flow is not ergodic on the whole bundle. The two examples are the extremities of the theory: the integrable flow of the torus and the periodic flow of the sphere are the classes in which the geodesic flow has no chaotic behaviour.

Jacobi Fields and Curvature

Definition. Let $\gamma$ be a geodesic and let $\gamma_s$ be a variation of $\gamma$ through geodesics with variation field $J$; the Jacobi equation along $\gamma$ is

$$ J''+R(J,\dot\gamma)\dot\gamma=0, $$

where $''$ is the covariant derivative along $\gamma$ and $R$ the curvature tensor; the solutions $J$ are the Jacobi fields, they span the tangent space at each point in the sense that every $J$ is determined by $J(0)$ and $J'(0)$, and the derivative of the geodesic flow along a variation of the initial condition is exactly the Jacobi field. A point $\gamma(t_0)$ with $t_0>0$ is conjugate to $\gamma(0)$ if there is a nonzero Jacobi field vanishing at both ends.

Theorem (curvature and the behaviour of the Jacobi fields). (i) If the sectional curvature of $M$ satisfies $K \le-k^2<0$, then the Jacobi field with $J(0)=0$, $\|J'(0)\|=1$ satisfies $\|J(t)\|\ge\frac{\sinh(kt)}{k}$ for $t>0$, and the Jacobi fields orthogonal to $\dot\gamma$ split into a family growing like $e^{kt}$ and a family decaying like $e^{-kt}$, the two being separated by a uniform exponential; consequently $M$ has no conjugate points.

(ii) If $K \ge k^2>0$, then the Jacobi fields of the normal directions vanish again before $\frac{\pi}{k}$ and $M$ has conjugate points; the exponential map is a local diffeomorphism up to the first conjugate point and no further.

Consequently, on a manifold of negative curvature, the geodesic flow expands the Jacobi fields of one family and contracts those of another at an exponential uniform rate, and this is the source of the hyperbolicity of the next section. The Jacobi fields, the conjugate points, the index form and the comparison theorems are those of Curvature and Geodesics.

Hyperbolicity of the Flow

Theorem (Anosov). Let $M$ be a compact Riemannian manifold of negative sectional curvature. Then the geodesic flow $\varphi_t$ on $SM$ is an Anosov flow: the tangent bundle splits as

$$ T_vSM=E^s_v\oplus E^0_v\oplus E^u_v , \qquad E^0_v=\mathbb{R}X(v) , $$

with $X$ the spray, and there are constants $C,\lambda>0$ such that $\|D\varphi_t|_{E^s}\|\le Ce^{-\lambda t}$ and $\|D\varphi_{-t}|_{E^u}\|\le Ce^{-\lambda t}$ for $t \ge0$; the strongly stable and strongly unstable distributions $E^s,E^u$ are integrable, and their integral manifolds are the horospheres: the leaf of $E^s$ through $v$ is the set of the unit vectors based at the points of the horosphere through $\pi(v)$ centred at the endpoint $\gamma_v(+\infty)$, and the leaf of $E^u$ is the analogous set for the horosphere centred at $\gamma_v(-\infty)$.

Proof (sketch). A vector of $T_vSM$ is a Jacobi field along $\gamma_v$ with $J(0)$ and $J'(0)$ orthogonal to the velocity; the component perpendicular to $\dot\gamma_v$ splits into the part whose norm decays exponentially and the part that grows, by the estimate of the previous section, uniformly in $v$ by the compactness and the strict negativity of the curvature. The integral manifolds are the level sets of the Busemann functions, which are the horospheres.

Corollary (structural stability and mixing). The geodesic flow of a compact negatively curved manifold is structurally stable, topologically transitive and topologically mixing, has a dense set of periodic orbits, and its periodic orbits are the closed geodesics; the entropy and the invariant measures are those of the hyperbolic theory of Hyperbolic Dynamics and Anosov Systems, of which the flow is the classical example, and its suspension structure is that of Smooth Dynamical Systems.

The Liouville Measure and the Ergodic Theorems

The Hopf Argument

Theorem (Hopf; Anosov–Sinai). Let $M$ be a compact Riemannian manifold of negative sectional curvature. Then the geodesic flow is ergodic with respect to the Liouville measure on $SM$: the only invariant measurable sets have measure $0$ or full measure, and consequently for almost every $v$ the orbit is equidistributed in $SM$.

Proof (sketch; the Hopf argument). Let $f$ be an invariant $L^2$ function and let $f^*(v)=\lim_T\frac1T\int_0^Tf(\varphi_tv)\,dt$ be the limit, which exists almost everywhere by the Birkhoff ergodic theorem. The limit $f^*$ is constant along the orbits, and it is constant along the leaves of the stable and the unstable foliations because the flow contracts the distances along those leaves and $f^*$ is invariant; the stable and unstable foliations are absolutely continuous, so their holonomy preserves the Liouville measure up to a bounded density, and the standard propagation argument — moving along the leaves, together with the local accessibility of the flow — shows that $f^*$ is constant almost everywhere. Hence every invariant $L^2$ function is constant and the flow is ergodic.

Theorem (mixing; Hedlund, Ratner, Dolgopyat). (i) For a compact manifold of constant negative curvature the horocycle flow is uniquely ergodic (Hedlund) and mixing, and the geodesic flow is mixing by the Howe–Moore theorem on the matrix coefficients of the semisimple group.

(ii) The geodesic flow of a compact negatively curved surface is mixing, with a rate of decay of correlations which is exponential for $C^r$ functions when the curvature is sufficiently regular, by the theorem of Dolgopyat; the exponential mixing, the equidistribution of the periodic orbits and the limit theorems for the geodesic excursions follow from the spectral gap of the transfer operator of the flow.

(iii) The Birkhoff averages converge to the Liouville average, and the fluctuations are governed by the central limit theorem for the Anosov flows, with a variance given by the Green–Kubo formula and the period function.

Entropy and the Volume Growth

Theorem (Pesin's formula and the entropy of the flow). Let $M$ be compact and negatively curved, and let $\mu$ be the Liouville measure on $SM$. Then

$$ h_\mu(\varphi_1)=\int_{SM}\sum_i\lambda_i^+\,d\mu , $$

the sum of the positive Lyapunov exponents of the time-one map, by Pesin's formula of Hyperbolic Dynamics and Anosov Systems; the exponents are the rates of expansion of the Jacobi fields, so that for a surface with curvature $-k^2$ they are $\pm k$ on the unit tangent bundle and the entropy per unit time is $k$.

Theorem (Dinaburg–Manning; the volume entropy). Let $M$ be a compact Riemannian manifold of nonpositive curvature with universal cover $\widetilde M$ and let $\mathrm{vol}$ be the volume of $\widetilde M$. Then the topological entropy of the geodesic flow is the volume entropy

$$ h_{\mathrm{top}}(\varphi)=\lim_{T\to\infty}\frac1T\log\mathrm{vol}(B(\tilde x,T)), $$

which is independent of $\tilde x$; for a manifold with curvature in the interval $[-a^2,-b^2]$, $a \ge b>0$, the entropy satisfies $(n-1)b\le h_{\mathrm{top}}\le(n-1)a$, with $n=\dim M$, and the entropy is $(n-1)k$ for the space form of constant curvature $-k^2$. For a compact hyperbolic surface of curvature $-1$ the entropy per unit time is $1$, and the Liouville measure is the measure of maximal entropy.

Proof (sketch). The volume of the ball of radius $T$ in the universal cover grows at the same exponential rate as the number of the $T$-separated points in $SM$, which is the topological entropy by the Bowen definition of Topological Dynamics; the comparison with the space forms gives the bounds, since the Jacobi field estimates of the previous sections compare the volume growth with the constant curvature models.

Periodic Orbits, the Length Spectrum and the Zeta Functions

Definition. The length spectrum of a compact Riemannian manifold is the set of the lengths of its closed geodesics, counted with multiplicity; a closed geodesic is the projection of a periodic orbit of the geodesic flow, and its length is the period. The prime geodesic theorem is the asymptotic count of the closed geodesics by length.

Theorem (prime geodesic theorem; Huber, Selberg). Let $M$ be a compact hyperbolic surface of curvature $-1$. Then the number of the closed geodesics of length at most $T$ satisfies

$$ \pi(T)=\#\{\gamma \text{ closed geodesic}:\ell(\gamma)\le T\}\sim\frac{e^{T}}{T} \qquad (T\to\infty), $$

the exponent $1$ being the topological entropy of the flow; for a compact negatively curved manifold of dimension $n$ and entropy $h$ the corresponding count grows like $e^{hT}/(hT)$, and the closed geodesics equidistribute in $SM$ with respect to the Liouville measure, by the equidistribution theorem for the periodic orbits of an Anosov flow.

Theorem (zeta functions). The Ruelle zeta function of the geodesic flow,

$$ \zeta_\varphi(s)=\prod_{\gamma}\bigl(1-e^{-s\ell(\gamma)}\bigr)^{-1}, $$

the product being over the primitive closed geodesics, converges for $\operatorname{Re}s>h_{\mathrm{top}}$ and extends meromorphically to the plane, with the poles and the zeros determined by the spectrum of the flow's transfer operator; for a compact hyperbolic surface it is the quotient $\zeta_\varphi(s)=Z(s+1)/Z(s)$ of two shifts of the Selberg zeta function $Z$, the telescoping $\prod_{k\ge0}(1-e^{-(s+1+k)\ell})/\prod_{k\ge0}(1-e^{-(s+k)\ell})=(1-e^{-s\ell})^{-1}$ turning the double product into the single one; the zeros and the poles of $Z$ are located at the spectral parameters of the Laplacian, and its functional equation reflects the symmetry $s\mapsto1-s$. The trace formula that relates the length spectrum to the spectrum of the Laplacian is the Selberg trace formula, and it is the arithmetic input of the counting of the closed geodesics; the spectral theory of the Laplacian and the trace formulae belong to the analysis of the operators of Unbounded Operators and Spectral Measures and to the analytic number theory of the corpus.

Example (the modular surface). For the noncompact modular surface $\mathbb{H}/SL(2,\mathbb{Z})$ the closed geodesics correspond to the periodic continued fractions, their lengths are determined by the trace of the corresponding hyperbolic element through $\cosh\tfrac{\ell}{2}=\tfrac12|\operatorname{tr}\gamma|$, the asymptotic form $\ell\sim2\log|\operatorname{tr}\gamma|$ holding as the trace grows, and the counting of the closed geodesics of length at most $T$ has the same main term $e^T/T$ with a logarithmic correction coming from the cusp; the coding of the flow by the continued fractions, the Gauss map as return map and the Lévy constant as mean return time are the subject of Dynamics and Number Theory.

Homogeneous Dynamics, Rigidity and the Nonpositive Case

Locally Symmetric Spaces

Theorem (homogeneous geodesic flows). Let $G$ be a connected semisimple Lie group of real rank one with finite centre, $K$ a maximal compact subgroup and $\GammaHomogeneous Dynamics, Ratner's Theorems and Equidistribution, and the lattices and the symmetric spaces are those of Lattices in Lie Groups and Riemannian Geometry.

Rigidity

Theorem (Mostow rigidity). Let $M$ and $N$ be closed hyperbolic manifolds of dimension $n \ge3$ with isomorphic fundamental groups. Then $M$ and $N$ are isometric. Equivalently, the geodesic flows of two closed hyperbolic manifolds of dimension $n \ge3$ are conjugate as flows if and only if the manifolds are isometric, so that the dynamics of the flow determines the geometry.

Theorem (entropy rigidity; Katok, Besson–Courtois–Gallot). Let $M$ be a closed Riemannian manifold of negative curvature and dimension $n$, with entropy per unit time $h_{\mathrm{top}}$ and volume $\operatorname{vol}(M,g)$. The scale-invariant product $h_{\mathrm{top}}^n\operatorname{vol}(M,g)$ is minimal exactly when $g$ has constant curvature $-1$; for a closed surface of genus $g\ge2$ the inequality is $h_{\mathrm{top}}^2\operatorname{vol}(M,g)\ge2\pi|\chi(M)|$, by Katok, with equality exactly for the metrics of constant curvature $-1$, and in higher dimensions the corresponding rigidity is that of Besson–Courtois–Gallot. The entropy alone carries no such bound: scaling the metric by $\lambda^2$ scales the lengths by $\lambda$ and the entropy by $\lambda^{-1}$, so that $h_{\mathrm{top}}$ has no positive lower bound and the rigidity is a statement about the normalised product. Consequently the entropy, which is a purely dynamical invariant of the geodesic flow, detects the constant curvature among the negatively curved metrics of a given volume.

The Nonpositive Case and the Open Problems

Remark (regularity of the hyperbolicity). If the curvature is nonpositive but vanishes somewhere, the geodesic flow is no longer uniformly hyperbolic and the flow is not Anosov; the invariant distributions still exist by the Pesin theory, the Lyapunov exponents are nonnegative, and the ergodicity of the flow on a compact surface of nonpositive curvature, with no focal points and with the flat points controlled, is a theorem of Pesin and of Ballmann and Burns–Gerber. The general problem of deciding which manifolds without conjugate points have an ergodic geodesic flow, and the extent to which the ergodicity fails at the flat points, is the open frontier of the theory; the nonuniform hyperbolicity and the invariant manifolds that are used are those of Hyperbolic Dynamics and Anosov Systems and Random Dynamical Systems, and the geometric hypotheses are those of Curvature and Geodesics.

Example (the sphere and the torus revisited). The round sphere has a completely periodic geodesic flow, its entropy is zero, and its invariant measures are the rotation-invariant ones; the flat torus has an integrable flow whose ergodic decomposition is by the direction of the velocity, the flow on the torus of a fixed irrational direction being ergodic and that on a rational direction being periodic. The contrast with the negative curvature — where the flow is ergodic and mixing and its entropy is positive — is the sharpest illustration of the role of the curvature in the ergodic theory of the flow, and it is the reason why the geodesic flow of the negative curvature is the model example of the hyperbolic dynamics.

Summary

The geodesic flow $\varphi_t(v)=\dot\gamma_v(t)$ on the unit tangent bundle $SM$ of a Riemannian manifold is the flow of the geodesic spray $\dot x^k=\xi^k$, $\dot\xi^k=-\Gamma^k_{ij}\xi^i\xi^j$, equivalently the Hamiltonian flow of $\frac12\|p\|^2$ on $T^*M$ restricted to $H=\frac12$; it preserves the Liouville measure (the Sasaki volume) and the contact form. The Jacobi equation $J''+R(J,\dot\gamma)\dot\gamma=0$ governs the derivative of the flow, and negative sectional curvature forces the Jacobi fields to grow and decay exponentially and excludes the conjugate points, which yields the theorem of Anosov that the geodesic flow of a compact negatively curved manifold is an Anosov flow with the splitting $E^s\oplus E^0\oplus E^u$, the stable and unstable manifolds being the horospheres. The flow is ergodic for the Liouville measure by the Hopf argument, which propagates the constancy of the ergodic averages along the absolutely continuous stable and unstable foliations, it is mixing (Hedlund, Ratner, Dolgopyat), the entropy of the Liouville measure is given by Pesin's formula $h_\mu=\int\sum\lambda_i^+$, while the topological entropy equals the volume entropy $\lim_T\frac1T\log\mathrm{vol}(B(\tilde x,T))$ by Dinaburg and Manning, the two agreeing for the space forms, with $(n-1)b\le h_{\mathrm{top}}\le(n-1)a$ for the curvature in $[-a^2,-b^2]$ and $h_{\mathrm{top}}=(n-1)k$ for the space form of curvature $-k^2$.

The closed geodesics are the periodic orbits of the flow; they satisfy the prime geodesic theorem $\pi(T)\sim e^{hT}/(hT)$, which for a compact hyperbolic surface is $e^T/T$, they equidistribute in $SM$, and they are counted by the poles of the Ruelle zeta function, which for the hyperbolic surfaces is the quotient $Z(s+1)/Z(s)$ of the Selberg zeta function $Z$, related to the spectrum of the Laplacian by the Selberg trace formula. For a locally symmetric space of rank one the geodesic flow is a homogeneous flow and the equidistribution, the orbit closures and the counting are those of Homogeneous Dynamics and Ratner's Theorems; Mostow rigidity and the entropy rigidity of Besson–Courtois–Gallot recover the geometry from the dynamics of the flow. When the curvature is nonpositive but degenerate the flow is only nonuniformly hyperbolic and the ergodicity is the subject of the Pesin theory and of the open problems of the field; the flat torus, whose flow is integrable with the ergodic decomposition by direction, and the round sphere, whose flow is periodic with zero entropy, are the extremities against which the negative curvature is measured.

Summary of Notation

Symbol	Meaning
$(M,g)$, $n=\dim M$	Riemannian manifold and its dimension
$TM$, $T^*M$, $SM$	tangent, cotangent and unit tangent bundles
$G$, $X$	geodesic spray and its vector field
$\varphi_t(v)=\dot\gamma_v(t)$	geodesic flow
$\Gamma^k_{ij}$	Christoffel symbols
$R$, $K$	curvature tensor, sectional curvature
$J$, $J''+R(J,\dot\gamma)\dot\gamma=0$	Jacobi field and Jacobi equation
$H(q,p)=\frac12\\|p\\|^2$	Hamiltonian of the kinetic energy
$\mu$	Liouville measure, the invariant measure of the flow
$E^s,E^0,E^u$	stable, flow, unstable distributions
$\lambda_i$	Lyapunov exponents of the flow
$h_{\mathrm{top}}$, $h_\mu$	topological and measure-theoretic entropy
$\ell(\gamma)$, $\pi(T)$	length of a closed geodesic, counting function
$\zeta_\varphi$, $Z$	Ruelle and Selberg zeta functions
$\Gamma\backslash G/K$	locally symmetric space of rank one

The Geodesic Flow

Introduction

The Geodesic Flow

The Flow of the Geodesic Spray

Jacobi Fields and Curvature

Hyperbolicity of the Flow

The Liouville Measure and the Ergodic Theorems

The Hopf Argument

Entropy and the Volume Growth

Periodic Orbits, the Length Spectrum and the Zeta Functions

Homogeneous Dynamics, Rigidity and the Nonpositive Case

Locally Symmetric Spaces

Rigidity

The Nonpositive Case and the Open Problems

Summary

Summary of Notation

Further Reading