---
tag: YZ4268SK8G
kind: statement
role: theorem
name: Morse index theorem for geodesics
msc: ["58E10", 53C22]
uses: [7CBT4SM83Y, NSYMQMHPDN, 9AHMM41PJ3, ZPXE5VSZ35, Q12AB1MJ64, MAX2ZYG6H9]
about: [ZPXE5VSZ35, MAX2ZYG6H9]
sources:
  - url: https://en.wikipedia.org/wiki/Conjugate_points
---

## Statement

Let $(M, g)$ be a [[Riemannian manifold|7CBT4SM83Y]] and let $\gamma \colon [0, 1] \to M$ be a [[geodesic|NSYMQMHPDN]] 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|Q12AB1MJ64]] 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:

1. The [[Morse index|MAX2ZYG6H9]] of $I$, the largest dimension of a subspace of $V$ on which $I$ is negative definite, is finite.
2. It equals the number of parameters $t \in (0, 1)$ such that $\gamma(t)$ is [[conjugate|ZPXE5VSZ35]] to $\gamma(0)$ along $\gamma$, each counted with its multiplicity.
3. 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|9AHMM41PJ3]] 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.

## Explanations

### L1 human

A geodesic is a critical point of the energy among paths with the same ends, but it need not be a minimum.
The index counts the independent ways of shortening it by a small deformation.
The theorem says that this number is the number of conjugate points along the way, where nearby geodesics from the start refocus.
On a sphere, a great circle arc longer than half the circle passes the antipode of its start, and it can be shortened by sliding it over the sphere.

### L2 human

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.