nLab Morse function

Definition

Consider a real-valued smooth function

(1)$$f \colon M \longrightarrow \mathbb{R}$$

on a smooth manifold $M$.

Definition

A point $p\in M$ is a critical point of $f$ (1) if for any smooth curve $\gamma : (-\epsilon, \epsilon)\to M$ with $\gamma(0)=p$, the tangent vector

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

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

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

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

Definition

The function $f$ is called a Morse function if every critical point of $f$ is regular (Def. ).

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.

Properties

One of the basic tools of Morse theory is the Morse lemma.

Related concepts

• perfect Morse function

References

See also:

• Wikipedia, Morse theory