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I recently learned from a knowledgeable person that for a Morse function $f: M \to R$ with a critical point $x_0$, one can perturb $f$ in such a fashion that the new function has the same critical points and the Hessian at $x_0$ can be arbitrarily arranged if the index is unchanged.

I have managed to prove this result, but it would be nice to know a textbook where this result can be found, since I need to refer to it. I think people who know more Morse theory than I do can help out.

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A stronger result--such a perturbation can be achieved by composing with an isotopy of the manifold--can be deduced from the Morse Lemma (which should be easy to find in the literature, since it is venerable and has a name). – Tom Goodwillie Jul 12 '12 at 17:40

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