# 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 $\gamma :\left(-ϵ,ϵ\right)\to M$ with $\gamma \left(0\right)=p$, the vector

$\frac{d\left(f\circ \gamma \right)}{\mathrm{dt}}{\mid }_{t=0}=0.$\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 $\varphi \left(p\right)=0\in {ℝ}^{n}$, the Hessian matrix

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