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Morse lemma

Let MM be a smooth manifold, f:Mf: M\to \mathbb{R} a real valued function.

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) 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 pp of ff there is a chart ϕ:U n\phi: U\to \mathbb{R}^n around pp 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 kk 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 AA such that in suitable local coordinates, quadratic functional fϕ 1f\circ\phi^{-1} can be written as f(p)+f(p)+<Ax,xA x,x>.

  • V. Guillemin, S. Sternberg, Geometric asymptotics, Appendix 1
Revised on March 9, 2010 19:28:18 by Zoran Škoda (161.53.130.104)