# Morse lemma

Let $M$ be a smooth $n$-dimensional manifold, and $f:M\rightarrow\mathbb{\mathbb{R}}$ a smooth 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 $f_{**}:T_{p}M\times T_{p}M\rightarrow\mathbb{R}$ (or $(f_{**})_{p}$ if $p$ need to be specified) the bilinear map

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

where $\tilde{v},\tilde{w}\in\mathcal{T}(M)$ are smooth 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 $x^{1},...,x^{n}$ on a neighborhood $U$ of $p$ we have

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

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

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

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

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

###### Theorem 1 (Morse lemma)

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

 $f|_{U}=f(p)-(x^{1})^{2}-...-(x^{\lambda})^{2}+(x^{\lambda+1})^{2}+...+(x^{n})^% {2},$

where $\lambda=\mathrm{Index}((f_{**})_{p})$.

Title Morse lemma MorseLemma 2013-03-22 13:53:12 2013-03-22 13:53:12 matte (1858) matte (1858) 18 matte (1858) Theorem msc 58E05 non degenerate critical point index of a bilinear map