nLab Morse lemma

Let MM be a smooth manifold and 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.

The 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 the number kk is determined by the Hessian matrix. While the Morse lemma is proved by Morse, the modern proof is by 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, the 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

Last revised on November 18, 2022 at 13:48:11. See the history of this page for a list of all contributions to it.