nLab
Morse function

A smooth real valued function $f:M\to ℝ$ on a smooth manifold $M$ is called a Morse function if every critical point of $f$ is regular.

$p\in M$ is a critical point of $f$ if for any curve $\gamma :(-ϵ,ϵ)\to M$ with $\gamma (0)=p$, the vector

$$\frac{d(f\circ \gamma )}{\mathrm{dt}}{\mid }_{t=0}=0.$$\frac{d(f\circ\gamma)}{dt} |_{t=0} = 0.

The critical point is regular if for one (or equivalently any) chart $\varphi :U^{open}\to {ℝ}^n$, where $p\in U$ and $\varphi (p)=0\in {ℝ}^n$, the Hessian matrix

$${(\frac{{\partial }^2(f\circ {\varphi }^{-1})}{\partial x^i\partial x^j}(0))}_{i,j=1,\dots ,n}$$\left(\frac{\partial^2 (f\circ \phi^{-1})}{\partial x^i\partial x^j}(0)\right)_{i,j=1,\ldots, n}

is a nondegenerate (i.e. maximal rank) matrix.

A choice of a Morse function on a compact manifold is often used to study topology of the manifold. This is called the Morse theory.

One of the basic tools is Morse lemma.

See also perfect Morse function and wikipedia: Morse theory