---
tag: AH059Y44NF
kind: definition
name: Morse index
msc: [57R70, "58E05", "58E10"]
defines: MAX2ZYG6H9
uses: [F45VQ47GPV, 0V33BNKZ9S, 0YTCMFB4HC, MGPPCDFQDG, 0AXZNE5XNH]
sources:
  - url: https://en.wikipedia.org/wiki/Morse_theory
---

## Definition

Let $M$ be a [[smooth manifold|F45VQ47GPV]], let $f \colon M \to \R$ be a [[smooth function|0V33BNKZ9S]], and let $p$ be a [[critical point|0YTCMFB4HC]] of $f$.
Let $H = \operatorname{Hess}_p f$ be its Hessian, the [[symmetric bilinear form|0AXZNE5XNH]] on $T_p M$ from the definition of a [[nondegenerate critical point|MGPPCDFQDG]].
The *Morse index* of $f$ at $p$ is the largest dimension of a linear subspace $V \subseteq T_p M$ such that $H(v, v) < 0$ for every nonzero $v \in V$.
The *nullity* of $f$ at $p$ is the dimension of the kernel $\set{v \in T_p M : H(v, w) = 0 \text{ for all } w \in T_p M}$.

## Remarks

In a chart, the index is the number of negative eigenvalues of the Hessian matrix, counted with multiplicity, and the nullity is the dimension of its kernel.
By Sylvester's law of inertia both are independent of the chart.
The point $p$ is nondegenerate exactly when the nullity is $0$, and the term Morse index is mostly used in that case.

Example: $f(x, y, z) = x^2 - y^2 - 3z^2$ on $\R^3$ has Hessian matrix $\operatorname{diag}(2, -2, -6)$ at the origin.
It is negative definite on the $yz$-plane and not on any $3$-dimensional subspace, so the index is $2$ and the nullity is $0$.

Non-example: for $g(x) = x^3$ on $\R$, the critical point $0$ has index $0$ and nullity $1$.
Index $0$ does not make $0$ a local minimum here, since the point is degenerate; this is why the index describes the local shape only at nondegenerate critical points.

The same definition applies on a Hilbert manifold, where the index may be infinite.
For the [[energy functional on paths|Q12AB1MJ64]], the index at a geodesic is computed by the [[Morse index theorem for geodesics|YZ4268SK8G]].