Morse lemma

Remarks

The idea of the proof is this. Work in a chart centred at $p$ with $f(p) = 0$; by Taylor's formula with integral remainder, $f(x) = \sum_{i,j} x_i x_j h_{ij}(x)$ with $h_{ij}$ smooth, symmetric in $i, j$, and $h_{ij}(0) = \frac{1}{2} \partial^2 f / \partial x_i \partial x_j(0)$. Since the matrix $(h_{ij}(0))$ is invertible, one can complete squares one variable at a time, as in the diagonalisation of a quadratic form, with smooth coefficients near $0$. Sylvester's law of inertia shows that the number of minus signs is the index of the Hessian. Example: $f(x, y) = x^2 - y^2 + y^3$ has a nondegenerate critical point of index $1$ at the origin, and near it the change $u = x$, $v = y \sqrt{1 - y}$ gives $f = u^2 - v^2$. Nondegeneracy is needed: $f(x) = x^3$ cannot be written as $\pm x_1^2$ in any chart near $0$, since it changes sign at $0$.

Going deeper

For a student

The Morse lemma is the local input of Morse theory. It shows that nondegenerate critical points are isolated, and it describes how the sublevel set $\set{f \le c}$ changes when $c$ passes a critical value: a handle of index $\lambda$ is attached. This leads to handle decompositions, to the Morse inequalities, and to Morse homology. It is the simplest case of the classification of singularities of smooth functions.