Global analysis, analysis on manifolds Critical point theory and variational problems in infinite dimensions Theorem

Morse index theorem for geodesics

Builds on Riemannian manifold Geodesic Jacobi field Conjugate points Energy functional on paths Morse index

Statement

Let $(M, g)$ be a Riemannian manifold and let $\gamma \colon [0, 1] \to M$ be a geodesic from $p = \gamma(0)$ to $q = \gamma(1)$ . Let $V$ be the vector space of continuous, piecewise smooth vector fields $W$ along $\gamma$ with $W(0) = 0$ and $W(1) = 0$ . Let $I$ be the Hessian at $\gamma$ of the energy functional on paths from $p$ to $q$ . It is the symmetric bilinear form on $V$ given by $I(W_1, W_2) = \int_0^1 \Big( g\big(\tfrac{D W_1}{dt}, \tfrac{D W_2}{dt}\big) - g\big(R(W_1, \dot\gamma) \dot\gamma, W_2\big) \Big) \, dt.$ Here $D/dt$ is the covariant derivative along $\gamma$ and $R$ is the Riemann curvature tensor. The sign is such that, for $W \perp \dot\gamma$ , $g(R(W, \dot\gamma)\dot\gamma, W)$ is $\abs{W}^2 \abs{\dot\gamma}^2$ times the sectional curvature of the plane of $W$ and $\dot\gamma$ . Then:

The Morse index of $I$ , the largest dimension of a subspace of $V$ on which $I$ is negative definite, is finite.
It equals the number of parameters $t \in (0, 1)$ such that $\gamma(t)$ is conjugate to $\gamma(0)$ along $\gamma$ , each counted with its multiplicity.
The nullity of $I$ equals the multiplicity of $q$ as a conjugate point of $p$ along $\gamma$ . It is $0$ if $q$ is not conjugate to $p$ .

Remarks

Integration by parts gives $I(W_1, W_2) = -\int_0^1 g\big(\tfrac{D^2 W_1}{dt^2} + R(W_1, \dot\gamma)\dot\gamma, W_2\big)\,dt$ plus terms at the breaks of $W_1$ , so the null space of $I$ consists of the Jacobi fields vanishing at both ends. For the index, one splits $[0, 1]$ into short intervals on which $\gamma$ is minimizing. Then $V$ decomposes into the finite-dimensional space of broken Jacobi fields and a complement on which $I$ is positive definite. Then one tracks how the index of $I$ restricted to $[0, \tau]$ changes as $\tau$ grows: it jumps exactly at conjugate points, by their multiplicity.

Example: on the unit sphere $S^n$ , a unit-speed great circle arc of length $L$ with $\pi < L < 2\pi$ passes one conjugate point, the antipode of its start, with multiplicity $n - 1$ . So its index is $n - 1$ , and for $n \ge 2$ it is not a local minimum of the energy. In particular, a geodesic that passes a conjugate point in its interior is not locally length-minimizing.

Going deeper

For a student

The theorem turns a variational question into a question about an ordinary differential equation, the Jacobi equation. It is the key input of Morse theory on path spaces: together with the Palais–Smale condition, it computes the cells attached at each geodesic. For example, on a manifold of nonpositive sectional curvature there are no conjugate points, so every geodesic has index $0$ . Bott used it to compute indices of geodesics in compact Lie groups in his proof of periodicity.

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Conjugate points concept
Energy functional on paths concept
Geodesic concept
Jacobi field concept
Morse index concept
Riemannian manifold concept
Morse index theorem for geodesics statement

Tag	`YZ4268SK8G`
URI	`m2://YZ4268SK8G`
Kind	statement, theorem
Writing	W0 stub
MSC	58E10, 53C22
Status	This copy of the site has no build data. Run `m2 status YZ4268SK8G`.
About	Conjugate points, Morse index
Uses	Riemannian manifold, Geodesic, Jacobi field, Conjugate points, Energy functional on paths, Morse index