Morse function

A smooth real valued function f:Mf:M\to\mathbb{R} on a smooth manifold MM is called a Morse function if every critical point of ff is regular.

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

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

Revised on April 6, 2010 23:04:37 by Zoran Škoda (