Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 5 down vote accepted

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.

share|improve this answer
thank you Sir ! –  The Common Crane Jan 4 '13 at 4:44

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.