The Geodesic Flow Operator

Introduction

The geodesics of a Riemannian manifold assemble into a single flow. Each tangent vector determines a geodesic, and following it for time $t$ displaces the vector to the tangent vector of the same geodesic at the later time; the resulting map on the tangent bundle is the geodesic flow. It is the flow of one vector field, the geodesic spray, and it is read here as an operator: the flow is a one-parameter group of substitutions of the tangent bundle, its generator is the spray read as a derivation, its differential along the flow is governed by the Jacobi fields, and the Hamiltonian formulation on the cotangent bundle exhibits the spray as the Hamiltonian vector field of the kinetic energy.

The article develops the operator reading. It defines the geodesic spray and the geodesic flow on the tangent bundle and on the unit tangent bundle; it writes the Hamiltonian formulation, with the kinetic energy on the cotangent bundle and the Legendre transform joining the two pictures, and states the flow as the one-parameter group generated by the Hamiltonian derivation; it computes the differential of the flow and derives the Jacobi equation from it, so that the Jacobi fields are exactly the derivatives of the flow; it introduces the Jacobi operator along a geodesic and identifies the conjugate points with its zeros; and it states the conservation of the energy and the completeness of the flow.

The article assumes the metric, the Levi-Civita connection, the geodesics, the exponential map and the curvature of Curvature and Geodesics and Riemannian Geometry, and it cites Smooth Manifolds and Differential Geometry for the tangent and cotangent bundles and the flow of a vector field. The symplectic form and the Hamiltonian theory of the cotangent bundle are named as structures of later articles, Symplectic Geometry and the Hamiltonian dynamics of Part III, and are used only in the sense already available; the measure-theoretic and spectral properties of the flow operator — the invariance of the Liouville measure, its unitary realisation on a Hilbert space and the entropy of the flow — belong to Part III and to The Geodesic Flow of the synthesis of this Part, and are cited rather than developed. No physics is invoked.

The Geodesic Spray

Vector Fields on the Tangent Bundle

Let $(M, g)$ be a Riemannian manifold of dimension $n$, with Levi-Civita connection $\nabla$, and let $\pi : TM \to M$ be the tangent bundle. A point of $TM$ is written $(x, \xi)$ with $x \in M$ and $\xi \in T_xM$; a chart $x^1, \ldots, x^n$ of $M$ induces the chart $x^1, \ldots, x^n, \xi^1, \ldots, \xi^n$ of $TM$, in which a tangent vector of $TM$ is a pair of $n$-tuples. The vertical subspace at $(x, \xi)$ is $\ker d\pi_{(x,\xi)} \cong T_xM$, and the canonical vertical field is the vector field whose value at $(x,\xi)$ is the copy of $\xi$ in the vertical subspace; it is written $V$ or, in the chart, $\xi^k\partial_{\xi^k}$.

Definition. The geodesic spray of $(M, g)$ is the vector field $G$ on $TM$ whose integral curves are the derivatives of the geodesics:

$$ G(x, \xi) = \frac{d}{dt}\Big|_{t=0}\bigl(\gamma_\xi(t), \gamma_\xi'(t)\bigr), $$

where $\gamma_\xi$ is the geodesic with $\gamma_\xi(0) = x$ and $\gamma_\xi'(0) = \xi$. In the induced chart it is

$$ G = \sum_k \xi^k\,\partial_{x^k} - \sum_{i,j,k} \Gamma^k_{ij}(x)\,\xi^i\xi^j\,\partial_{\xi^k}, $$

with the Christoffel symbols $\Gamma^k_{ij}$ of the metric. The geodesic flow is the flow of $G$,

$$ \varphi_t : TM \longrightarrow TM, \qquad \varphi_t(x, \xi) = \bigl(\gamma_\xi(t), \gamma_\xi'(t)\bigr), $$

defined for those $t$ for which the geodesic is defined, and $\varphi_{s+t} = \varphi_s \circ \varphi_t$ wherever both sides are defined.

Proposition. The spray is a second-order vector field, in the sense that $d\pi(G(x,\xi)) = \xi$; conversely a second-order vector field on $TM$ whose integral curves are the derivatives of curves on $M$ determines a spray exactly when the curves are the geodesics of a metric, and then the field is $G$ above. The geodesic equation is the integral equation of the spray, $\ddot x^k + \sum_{ij}\Gamma^k_{ij}\dot x^i\dot x^j = 0$, and it is a second-order ordinary differential equation.

Proof. The first component of $G$ is $\xi$, which is $d\pi(G) = \xi$; the second is the negative of the Christoffel expression, which is exactly the geodesic equation. The local solvability and the smooth dependence on the initial condition are the theory of ordinary differential equations of Part III, cited; the geodesic equation is the equation of the spray.

Conservation of the Energy

Definition. The energy of a tangent vector is

$$ E(x, \xi) = \tfrac{1}{2}\,g_x(\xi, \xi), $$

and for $r > 0$ the sphere bundle of radius $r$ is $S_rM = \{E = \tfrac12 r^2\}$, with $SM = S_1M$ the unit tangent bundle.

Proposition. The energy is constant along the geodesic flow, $E \circ \varphi_t = E$; consequently the flow preserves every sphere bundle $S_rM$, and the flow restricted to $S_rM$ is the flow of the geodesic vector field $G_r$ obtained by projecting $G$ along the radius. A geodesic has constant speed, equal to the radius of the sphere bundle that contains its derivative.

Proof. A geodesic has constant speed, so $E(\varphi_t(x,\xi)) = \frac12|\gamma_\xi'|^2 = \frac12 g_x(\xi,\xi) = E(x,\xi)$; the level sets of $E$ are the sphere bundles, and $G$ is tangent to them, so it projects.

The Hamiltonian Formulation

The Cotangent Bundle and the Legendre Transform

Definition. The Legendre transform of the metric is the bundle isomorphism

$$ \flat : TM \longrightarrow T^*M, \qquad \flat(x, \xi) = \bigl(x,\ g_x(\xi, \cdot)\bigr), $$

with inverse $\sharp$, and the kinetic energy Hamiltonian is the function on $T^*M$

$$ H(x, p) = \tfrac{1}{2}\,g_x^{\sharp}(p, p), \qquad H\circ\flat = E . $$

In a chart, $p_i = \sum_j g_{ij}\xi^j$ and $H = \frac12\sum_{ij}g^{ij}(x)p_ip_j$.

The cotangent bundle carries the canonical symplectic form $\omega = \sum_i dp_i\wedge dx^i$, a nondegenerate closed $2$-form; the form is a structure of Part II and the pair $(T^*M, \omega)$ is the symplectic manifold of Symplectic Geometry, later in this Part, and of the symplectic topology of Part II. The Poisson bracket of two functions is $\{f, h\} = \omega(\sharp_\omega df, \sharp_\omega dh)$, with $\sharp_\omega$ the isomorphism $T^*T^*M \to TT^*M$ defined by $\omega$.

Theorem. The Legendre transform carries the geodesic spray to the Hamiltonian vector field of $H$: writing $X_H$ for the vector field with $\omega(X_H, \cdot) = -dH$, one has $d\flat(G) = X_H$, and the integral curves of $X_H$ are the curves $t \mapsto \flat(\varphi_t(x,\xi))$. Equivalently, the geodesic flow is the Hamiltonian flow of the kinetic energy on the cotangent bundle, and every integral curve satisfies the Hamilton equations

$$ \dot x^k = \frac{\partial H}{\partial p_k} = \sum_j g^{kj}p_j, \qquad \dot p_k = -\frac{\partial H}{\partial x^k} = -\tfrac{1}{2}\sum_{ij}\partial_{x^k}g^{ij}\,p_ip_j . $$

Proof. The two equations are the equations of the spray read through $\flat$: the first is $\dot x^k = \xi^k$, and the second is obtained from $\dot\xi^k = -\sum_{ij}\Gamma^k_{ij}\xi^i\xi^j$ by differentiating $p_k = \sum_jg_{kj}\xi^j$ and using the identity $\partial_{x^l}g_{kj} = \sum_i(\Gamma^i_{kl}g_{ij}+\Gamma^i_{jl}g_{ki})$, which expresses the metricity of $\nabla$ in a chart. The Hamiltonian field is well defined because $\omega$ is nondegenerate and closed, and it is tangent to the level sets of $H$ because $X_H(H) = \{H,H\} = 0$.

The Flow as a One-Parameter Group

Definition. The geodesic flow operator is the family $\{U_t\}$ acting on a function $f$ on $TM$ by substitution,

$$ (U_tf)(x, \xi) = f\bigl(\varphi_t(x, \xi)\bigr), $$

and on a function on $T^*M$ by the same formula along the Hamiltonian flow. The domain of each $U_t$ is the set on which $\varphi_t$ is defined, and $U_{s+t} = U_s\circ U_t$, $U_0 = \mathrm{id}$, wherever the compositions are defined.

Proposition. The generator of the group is the spray read as a derivation of the algebra of smooth functions,

$$ \frac{d}{dt}\Big|_{t=0} U_tf = Gf, \qquad \text{equivalently} \qquad \frac{d}{dt} U_tf = U_t(Gf), $$

and on the cotangent bundle the generator is the Hamiltonian derivation $\{H,\cdot\}$. The flow operator commutes with the energy, $U_tE = E$, and its restriction to the functions on $SM$ is again a one-parameter group.

Proof. The first identity is the chain rule for the flow of $G$, and the second follows by differentiating $U_{t+s} = U_tU_s$ at $s = 0$. The Hamiltonian generator is $\{H,\cdot\}$ by the definition of the Hamiltonian vector field, and the commutation with $E$ is the constancy of the energy along the flow.

Remark (the flow operator on a Hilbert space). The flow preserves the Liouville measure $\mu$ on $TM$ or on $SM$, the measure whose density is the Riemannian volume in the position variable and the Lebesgue measure in the velocity variable; once the measure and the integral are available, this makes each $U_t$ a unitary operator on the Hilbert space $L^2(SM, \mu)$. The measure, the unitary realisation and the spectral properties of the flow operator are Part III's, and the ergodic theory of the flow is The Geodesic Flow of the synthesis of this Part; both are cited and neither is developed here.

The Differential of the Flow and the Jacobi Fields

Jacobi Fields

Definition. Let $\gamma$ be a geodesic and let $J$ be a vector field along $\gamma$, that is a smooth map $t \mapsto J(t) \in T_{\gamma(t)}M$. It is a Jacobi field if it satisfies the Jacobi equation

$$ \frac{D^2J}{dt^2} + R\bigl(J, \gamma'\bigr)\gamma' = 0, $$

where $\frac{D}{dt}$ is the covariant derivative along $\gamma$ and $R$ the curvature tensor. The vector fields along $\gamma$ form a real vector space $\mathrm{X}(\gamma)$, and the Jacobi operator along $\gamma$ is the linear endomorphism

$$ \mathcal{J}_\gamma : \mathrm{X}(\gamma) \longrightarrow \mathrm{X}(\gamma), \qquad \mathcal{J}_\gamma J = \frac{D^2J}{dt^2} + R\bigl(J, \gamma'\bigr)\gamma' , $$

so that the Jacobi fields are the kernel of $\mathcal{J}_\gamma$.

Theorem. The Jacobi equation is the linearisation of the geodesic equation: a family $s \mapsto \gamma_s$ of geodesics with $\gamma_0 = \gamma$ has variation field $J(t) = \partial_s\gamma_s(t)|_{s=0}$, and $J$ is a Jacobi field; conversely every Jacobi field arises this way. Moreover the Jacobi fields along $\gamma$ are exactly the derivatives of the geodesic flow,

$$ J(t) = \bigl(\pi \circ d\varphi_t\bigr)(w), \qquad w \in T_{(x,\xi)}TM,\quad (x,\xi) = (\gamma(0),\gamma'(0)), $$

where $\pi : TTM \to TM$ is the bundle projection followed by the identification of the vertical subspaces with the tangent spaces.

Proof sketch. Differentiate the geodesic equation $\nabla_{\partial_t}\gamma_s' = 0$ in the variation parameter $s$ and commute the derivatives with the curvature formula $R(\partial_s,\partial_t)\partial_t = \nabla_{\partial_s}\nabla_{\partial_t}\partial_t - \nabla_{\partial_t}\nabla_{\partial_s}\partial_t$; the term $\nabla_{\partial_t}\nabla_{\partial_t}\partial_s$ gives $\frac{D^2J}{dt^2}$ and the remaining term is $R(J,\gamma')\gamma'$, which is the equation. For the converse, the map from the $(2n)$-dimensional space $T_{(x,\xi)}TM$ to the space of Jacobi fields along $\gamma$, $w\mapsto(\pi\circ d\varphi_t)(w)$, is well defined by the first part, and it is linear; it is surjective because a Jacobi field is determined by $(J(0), \frac{DJ}{dt}(0))$, a pair of vectors in $T_xM$, of total dimension $2n$, and the equation is second order, so $\dim\ker = 0$ and the map is an isomorphism. The identification of the two $2n$-dimensional spaces is the content of the Jacobi equation.

Conjugate Points

Definition. A point $\gamma(t_0)$ with $t_0 > 0$ is conjugate to $\gamma(0)$ along $\gamma$ if there is a nonzero Jacobi field $J$ along $\gamma$ with $J(0) = 0$ and $J(t_0) = 0$. The dimension of the space of such fields is the multiplicity of the conjugate point.

Theorem. The point $\gamma(t_0)$ is conjugate to $\gamma(0)$ if and only if $d(\exp_{\gamma(0)})_{t_0\gamma'(0)}$ is singular, and then the two maps have the same kernel dimension. A geodesic ceases to be minimising at the first conjugate point in the following sense: if there is a conjugate point at $t_0$ then $\gamma$ is not minimising on $[0, t_0 + \epsilon]$ for any $\epsilon > 0$, and a geodesic with no conjugate point before $t_1$ is minimising on $[0, t_1]$.

Proof sketch. The derivative of the exponential is computed from the Jacobi fields by $d(\exp_x)_v(w) = J_w(1)$, where $J_w$ is the Jacobi field with $J_w(0) = 0$, $\frac{DJ_w}{dt}(0) = w$; this is the Gauss lemma read through the flow and is the same computation as the theorem above. The singularity at $t_0$ is therefore the existence of a nonzero $w$ with $J_w(t_0) = 0$, which is a conjugate point. The minimising statement is the second-variation formula: at a conjugate point the second variation has a nonpositive direction, so minimality fails; and the absence of conjugate points makes the second variation positive definite, so minimality holds. The details are the classical index theory of Jacobi, and the analytic part belongs to Part III.

Corollary (the Jacobi operator and the second variation). The second variation of the energy at a geodesic $\gamma$ is the quadratic form $J \mapsto \int_0^1 \bigl(|\frac{DJ}{dt}|^2 - g(R(J,\gamma')\gamma', J)\bigr)\,dt$ on the variation fields vanishing at the endpoints, and its Euler–Lagrange equation is the Jacobi equation; equivalently the second variation is the pairing $\int_0^1 g(\mathcal{J}_\gamma J, J)\,dt$ with the suitable boundary conditions. With respect to the pairing $\int_0^1 g(\cdot,\cdot)\,dt$ the Jacobi operator is self-adjoint, so its index is the number of conjugate points counted with multiplicity.

Proof. Integrate $\int g(\frac{D^2J}{dt^2},J)$ by parts and use the antisymmetry of the curvature: $\int g(\frac{D^2J}{dt^2},J) = -\int|\frac{DJ}{dt}|^2$ for fields vanishing at the endpoints, and $g(R(J,\gamma')\gamma',J)$ is symmetric in the two slots of the curvature, so the form is the stated one and its Euler–Lagrange equation is the Jacobi equation. The self-adjointness of $\mathcal{J}_\gamma$ for the pairing is the same integration by parts, and the index statement is the standard correspondence between the negative directions and the conjugate points; the spectral theory of the self-adjoint operator is Part III's.

Completeness of the Flow

Theorem. The geodesic spray is a complete vector field if and only if $(M, g)$ is geodesically complete, and then the geodesic flow is defined for all real $t$; for a complete connected manifold every pair of points is joined by a minimising geodesic. A compact Riemannian manifold is complete, and so is a closed submanifold of Euclidean space with the induced metric.

Proof. The integral curves of $G$ are the derivatives of the geodesics, so completeness of $G$ is geodesic completeness by definition; the equivalence of geodesic completeness with metric completeness and with the existence of minimising geodesics is the theorem of Hopf–Rinow of Riemannian Geometry, cited. A compact metric space is complete, and a closed subset of a Euclidean space is complete, which gives the two examples.

Summary

The geodesic spray $G$ on $TM$ is the second-order vector field whose integral curves are the derivatives of the geodesics; in a chart it is $G = \xi^k\partial_{x^k} - \Gamma^k_{ij}\xi^i\xi^j\partial_{\xi^k}$, and its flow is the geodesic flow $\varphi_t$. Under the Legendre transform $\flat$ the spray becomes the Hamiltonian vector field of the kinetic energy $H = \frac12 g^{\sharp}(p,p)$ on the cotangent bundle, so the geodesic flow is the Hamiltonian flow of $H$, and the Hamilton equations are the geodesic equation read through $\flat$. The flow operator $U_tf = f\circ\varphi_t$ is a one-parameter group generated by the derivation $G$, equivalently by $\{H,\cdot\}$; it preserves the energy and every sphere bundle, and it preserves the Liouville measure, making it unitary on $L^2(SM,\mu)$ once the measure is available from Part III.

The derivative of the flow along a geodesic is the Jacobi field: the linearisation of the geodesic equation is $\frac{D^2J}{dt^2}+R(J,\gamma')\gamma' = 0$, and the Jacobi fields are exactly the images $\pi\circ d\varphi_t$ of tangent vectors of $TM$. The Jacobi operator $\mathcal{J}_\gamma J = \frac{D^2J}{dt^2}+R(J,\gamma')\gamma'$ has the Jacobi fields as its kernel, it is self-adjoint for the pairing $\int g(\cdot,\cdot)\,dt$, and its index counts the conjugate points: $\gamma(t_0)$ is conjugate to $\gamma(0)$ exactly when there is a nonzero Jacobi field vanishing at both ends, equivalently when $d\exp$ is singular there, and the first conjugate point is where the geodesic ceases to be minimising. The spray is complete exactly when the metric is geodesically complete, by Hopf–Rinow, and then the flow is defined for all time.

Summary of Notation

Symbol	Meaning
$(M, g)$, $n=\dim M$, $\nabla$, $R$	Riemannian manifold, connection and curvature tensor
$TM$, $T^*M$, $\pi$	Tangent and cotangent bundles and the projection
$(x, \xi)$, $(x, p)$	Points of $TM$ and $T^*M$; $p = \flat(\xi)$
$\Gamma^k_{ij}$	Christoffel symbols
$G$, $\varphi_t$	Geodesic spray and geodesic flow
$E(x,\xi) = \frac12g_x(\xi,\xi)$	Energy; constant along the flow
$S_rM$, $SM$	Sphere bundle of radius $r$; unit tangent bundle
$\flat$, $\sharp$	Legendre transform and its inverse
$H(x,p) = \frac12 g^{\sharp}_x(p,p)$	Kinetic energy Hamiltonian; $H\circ\flat=E$
$\omega = \sum_i dp_i\wedge dx^i$, $\{\cdot,\cdot\}$	Canonical symplectic form and Poisson bracket
$X_H$	Hamiltonian vector field, $\omega(X_H,\cdot) = -dH$
$U_t$, $U_tf = f\circ\varphi_t$	Geodesic flow operator and its generator
$J$, $\frac{D^2J}{dt^2}+R(J,\gamma')\gamma'=0$	Jacobi field and the Jacobi equation
$\mathcal{J}_\gamma$	Jacobi operator along $\gamma$; kernel the Jacobi fields
Conjugate point, multiplicity	Zeros of a Jacobi field; dimension of the space of such fields
Hopf–Rinow	Completeness $\iff$ geodesic completeness $\iff$ finite compactness