Morse lemma with least amount of regularity.

Question

I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across with work only for $C^3$ Morse functions.

A Google search was inconclusive about the existence of a Morse lemma for Morse functions $f: M \to \Bbb R$ with lesser regularity then $C^3$, where $M$ is a smooth finite dimensional manifold.

A reference is perhaps the best possible answer, but any chunk of information will be appreciated.

Answer 1

You only need $C^2$. See Nirenberg's book Topics in Nonlinear Functional Analysis, Theorem 3.1.1. He attributes this version of the Morse lemma to the late great Lars Hormander, Fourier Integral Operators I.

Answer 2

Requiring $C^2$ is too much: you can ask only $C^1$, twice differentiable at the distinguished point with a non-degenerate Hessian matrix. More precisely the following holds true.

Theorem. Let $\Omega$ be an open subset of $\mathbb R^n$, $x_{0}\in \Omega$ and $f:\Omega\longrightarrow \mathbb R$ be a $C^1$ function twice differentiable at $x_{0}$ such that $$ df(x_{0})=0,\quad\det f''(x_{0})\not=0. $$ Then there exist a neighborhood $V$ of $0$ in $\mathbb R^n$, a neighborhood $U$ of $x_{0}$ in $\Omega$, and a $C^1$ diffeomorphism $\kappa:V\longrightarrow U$ such that $\kappa(0)=x_{0},\kappa'(0)=Id,$ \begin{equation} (f\circ \kappa)(y)=f(x_{0})+\frac12\langle{ f''(x_{0})y},{y}\rangle. \end{equation}

For a proof, try your hand or check Santiago Lopez de Medrano, MR 1378414.