Morse lemma

Let $M$ be a smooth $n$-dimensional manifold, and $f:M\to ℝ$ asmooth map. We denote by $\mathrm{Crit}(f)$ the set of critical points of$f$, i.e.

$$\mathrm{Crit}(f)=\{p\in M|{(f_{*})}_p=0\}$$

For each $p\in \mathrm{Crit}(f)$ we denote by $f_{**}:T_{p}M\times T_{p}M\to ℝ$ (or ${(f_{**})}_p$ if $p$ need to be specified) thebilinear map

$$f_{**}(v,w)=v(\tilde{w}(f))=w(\tilde{v}(f)),\forall v,w\in T_{p}M,$$

where $\tilde{v},\tilde{w}\in 𝒯(M)$ aresmooth vector fields such that ${\tilde{v}}_p=v$ and ${\tilde{w}}_p=w$.This is a good definition. In fact $p\in \mathrm{Crit}(f)$ implies

$$v(\tilde{w}(f))-w(\tilde{v}(f))={(\tilde{v}(f),\tilde{w}(f))}_p=f_{*}{(\tilde{v},\tilde{w})}_p=0.$$

In smooth local coordinates$x^1,\mathrm{\dots },x^n$ on a neighborhood $U$ of $p$ we have

$$f_{**}({\frac{\partial }{\partial x^i}|}_p,{\frac{\partial }{\partial x^j}|}_p)=\frac{{\partial }^{2}f}{\partial x^i\partial x^j}(p).$$

A critical point $p\in \mathrm{Crit}(f)$ is called non degeneratewhen the matrix

$${\left(\frac{{\partial }^{2}f}{\partial x^i\partial x^j}(p)\right)}_{i,j\in \{1,\mathrm{\dots },n\}}$$

is non singular. We can equivalentlyexpress this condition without the use of local coordinates sayingthat $p\in \mathrm{Crit}(f)$ is non degenerate when for each $v\in T_{p}M\setminus \{0\}$ the linear functional $f_{**}(v,\cdot )\in \mathrm{Hom}(T_{p}M,ℝ)$is not zero, i.e. there exists $w$ such that $f_{**}(v,w)\ne 0$.

We recall that the index of a bilinear functional $H:V\times V\to ℝ$ is the dimension $\mathrm{Index}(H)$ of a maximal linearsubspace $W\subseteq V$ such that $H$ is negative definite on$W\times W$.

Theorem 1 (Morse lemma)

Let $f\mathrm{:}M\mathrm{\to }\mathrm{R}$ be a smooth map. For each non degenerate $p\mathrm{\in }\mathrm{Crit}\mathit{}\mathrm{(}f\mathrm{)}$ there exists a neighborhood $U$ of $p$ and smooth coordinates $x\mathrm{=}\mathrm{(}{x}^{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{x}^n\mathrm{)}$ on $U$ such that $x\mathit{}\mathrm{(}p\mathrm{)}\mathrm{=}\mathrm{0}$ and

$${f|}_U=f(p)-{(x^1)}^2-\mathrm{\dots }-{(x^{\lambda })}^2+{(x^{\lambda +1})}^2+\mathrm{\dots }+{(x^n)}^2,$$

where $\lambda \mathrm{=}\mathrm{Index}\mathit{}\mathrm{(}{\mathrm{(}{f}_{\mathrm{*}\mathrm{*}}\mathrm{)}}_p\mathrm{)}$.