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.


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


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


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


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


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


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