Morse lemma

Let M be a smooth manifold, f:M a real valued function.

pM is a critical point of f if for any curve γ:(ϵ,ϵ)M with γ(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, where pU and ϕ(p)=0 n, the Hessian matrix

( 2(fϕ 1)x ix j 0) ij\left(\frac{\partial^2 (f\circ \phi^{-1})}{\partial x^i\partial x^j}|_{0}\right)_{ij}

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

Morse lemma states that for any regular critical point p of f there is a chart ϕ:U n around p such that the function in these coordinates is quadratic:

(fϕ 1)(x 1,,x n)=f(p)+ i=1 kx i 2 j=k+1 nx j 2(f\circ\phi^{-1})(x^1,\ldots,x^n) = f(p) +\sum_{i=1}^k x_i^2 - \sum_{j=k+1}^n x_j^2

and number k is determined by the Hessian matrix. While the Morse lemma is proved by Morse, the modern proof is by the Moser’s deformation method. The Morse lemma can be generalized to smooth functions on a Hilbert manifold, in which case there is a linear operator A such that in suitable local coordinates, quadratic functional fϕ 1 can be written as f(p)+<Ax,x>.

  • V. Guillemin, S. Sternberg, Geometric asymptotics, Appendix 1
Revised on March 9, 2010 19:28:18 by Zoran Škoda (