nLab
Morse function

Contents

Contents

Definition

Consider a real-valued smooth function

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

on a smooth manifold MM.

Definition

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

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

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

( 2(fϕ 1)x ix j(0)) i,j=1,,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.

Definition

The function ff is called a Morse function if every critical point of ff 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.

References

See also:

Last revised on January 31, 2021 at 04:08:38. See the history of this page for a list of all contributions to it.