Morse function

Remarks

At a critical point the Hessian transforms under a change of chart by $H \mapsto A^T H A$ with $A$ invertible, so nondegeneracy and the index do not depend on the chart. Example: $f \colon \R^2 \to \R$, $f(x, y) = x^2 - y^2$, is a Morse function. Its only critical point is the origin, where the Hessian is $\begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix}$, which is invertible; the index is $1$. Non-example: $g \colon \R \to \R$, $g(x) = x^3$, is not a Morse function. Its critical point $0$ has Hessian $g''(0) = 0$, which is not invertible. Nondegenerate critical points are isolated, so on a compact manifold a Morse function has only finitely many critical points. On a compact manifold the Morse functions form an open dense subset of $C^\infty(M, \R)$.