Morse Function

Overview

A Morse function is a smooth height function with only nondegenerate critical points. Near each critical point its shape is controlled by a nonsingular quadratic form. Tracking sublevel sets then connects derivatives of a function to the topology of its domain. The critical points indicate possible topological changes, while their indices specify the dimensions of the cells involved.

Prerequisites

Use differentials of smooth functions, Hessian matrices, symmetric bilinear forms, and the inverse function theorem. Homology and CW complexes are needed for the global applications. A Riemannian metric helps construct a gradient flow, but the definition of a Morse function does not depend on choosing one.

Definition and conventions

Let $M$ be a smooth Hausdorff, second countable $n$-manifold without boundary. A point $p$ is critical for $f:M\to\mathbb R$ when $df_p=0$. At a critical point, the matrix

represents an intrinsic symmetric bilinear form on $T_pM$, the Hessian. Under a change of coordinates, extra chain-rule terms vanish because $df_p=0$; the matrices are related by congruence.

The critical point is nondegenerate when this form is nonsingular. Its index $\lambda(p)$ is the dimension of a maximal subspace on which the Hessian is negative definite. Equivalently, it counts negative eigenvalues in any coordinate representation. The function is Morse when every critical point is nondegenerate. A function with no critical points satisfies this condition vacuously. Distinct critical values are not required.

Intuition

A nondegenerate critical point has no infinitesimally flat direction in its quadratic approximation. A minimum has index zero; a maximum has index $n$; intermediate indices describe saddles. The Morse lemma strengthens the approximation: suitable smooth coordinates remove all higher-order terms locally, giving an exact quadratic normal form.

This local conclusion must be distinguished from global control. On a noncompact domain, topology can be influenced by behavior at infinity unless suitable compactness or properness conditions are supplied.

Examples and non-examples

On $\mathbb R^2$, $f(x,y)=x^2-3y^2$ has one critical point, with Hessian $\operatorname{diag}(2,-6)$ and index one. In contrast, $x^4+y^2$ has an isolated strict minimum at the origin but Hessian $\operatorname{diag}(0,2)$. Isolation and strict minimality do not imply the Morse condition.

On the torus $T^2=(\mathbb R/2\pi\mathbb Z)^2$, take

The derivative vanishes exactly when both angles are $0$ or $\pi$. Its Hessian there is $\operatorname{diag}(-\cos\theta,-2\cos\phi)$. The point $(\pi,\pi)$ has index zero, $(0,0)$ index two, and the other two points index one. Their values are respectively $-3,3,-1,1$. Consequently $c_0-c_1+c_2=1-2+1=0$, matching the torus's Euler characteristic. The formula is well defined on the quotient because cosine is periodic.

The height function $f(x)=x$ on $\mathbb R$ has no critical points and is Morse. A constant function on a positive-dimensional manifold is not: its Hessian vanishes at every point.

Important properties

The Morse lemma gives coordinates centered at a nondegenerate critical point with

Proof idea: Taylor's formula writes $f-f(p)$ as a quadratic expression with smooth coefficient functions. Nondegeneracy allows successive smooth completions of squares; the inverse function theorem verifies that the resulting transformations are coordinates. Sylvester's law of inertia fixes the number of negative squares.

For a Morse function on a closed manifold, critical points are isolated and form a closed set, hence there are finitely many. Between regular levels containing no critical value, the sublevel sets are diffeomorphic. The normalized gradient field $\nabla f/\|\nabla f\|^2$ changes the function value at unit rate; compactness controls its flow across the interval.

Crossing a single critical point of index $\lambda$ attaches a $\lambda$-cell up to homotopy. Thus a closed manifold has the homotopy type of a finite CW complex with one cell for each critical point. If $c_k$ counts index-$k$ points, then $\chi(M)=\sum_k(-1)^kc_k$ and $\dim H_k(M;\mathbb F)\leq c_k$ for a field $\mathbb F$. Orientability is unnecessary for these statements. Manifolds with boundary require additional boundary hypotheses.

Connections and further reading

Morse theory turns local calculus into global topology; energy functionals on path spaces lead toward geodesic applications with further analytic assumptions. Read Cohen–Iga–Norbury, Chapter 5 for the inspected normal form, sublevel-set arguments, and homological consequences.