Singularity theory

Morse lemma

Let b be a non-degenerate critical point of f : M → R. Then there exists a chart (x1, x2, ..., x_n) in a neighborhood U of b such that $\displaystyle x_{i}(b)=0$ for all i and

$f(x)=f(b)-x_{1}^{2}-\cdots -x_{\alpha }^{2}+x_{\alpha +1}^{2}+\cdots +x_{n}^{2}$

throughout U. Here α is equal to the index of f at b.

As a corollary of the Morse lemma, one sees that non-degenerate critical points are isolated. (Regarding an extension to the complex domain see Complex Morse Lemma. For a generalization, see Morse–Palais lemma).