Morse lemma


Let M be a smooth n-dimensional manifold, and f:M a smooth map. We denote by Crit(f) the set of critical points of f, i.e.

Crit(f)={pM|(f*)p=0}

For each pCrit(f) we denote by f**:TpM×TpM (or (f**)p if p need to be specified) the bilinear map

f**(v,w)=v(w~(f))=w(v~(f)),v,wTpM,

where v~,w~𝒯(M) are smooth vector fieldsMathworldPlanetmath such that v~p=v and w~p=w. This is a good definition. In fact pCrit(f) implies

v(w~(f))-w(v~(f))=(v~(f),w~(f))p=f*(v~,w~)p=0.

In smooth local coordinates x1,,xn on a neighborhoodMathworldPlanetmath U of p we have

f**(xi|p,xj|p)=2fxixj(p).

A critical point pCrit(f) is called non degenerate when the matrix

(2fxixj(p))i,j{1,,n}

is non singular. We can equivalently express this condition without the use of local coordinates saying that pCrit(f) is non degenerate when for each vTpM{0} the linear functionalMathworldPlanetmath f**(v,)Hom(TpM,) is not zero, i.e. there exists w such that f**(v,w)0.

We recall that the index of a bilinear functional H:V×V is the dimensionPlanetmathPlanetmathPlanetmath Index(H) of a maximal linear subspace WV such that H is negative definitePlanetmathPlanetmath on W×W.

Theorem 1 (Morse lemma)

Let f:MR be a smooth map. For each non degenerate pCrit(f) there exists a neighborhood U of p and smooth coordinatesPlanetmathPlanetmath x=(x1,,xn) on U such that x(p)=0 and

f|U=f(p)-(x1)2--(xλ)2+(xλ+1)2++(xn)2,

where λ=Index((f**)p).

Title Morse lemma
Canonical name MorseLemma
Date of creation 2013-03-22 13:53:12
Last modified on 2013-03-22 13:53:12
Owner matte (1858)
Last modified by matte (1858)
Numerical id 18
Author matte (1858)
Entry type Theorem
Classification msc 58E05
Defines non degenerate critical point
Defines index of a bilinear map