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Let $f:\mathbb{R}^{m+k}\mapsto\mathbb{R}^k$ be a smooth function. I have seen quite a few books for Morse theory for $f$ when $k=1$. Is there a generalization to $k\geq2$? When $k=1$, we can define a Morse function $f$ by checking the eigenvalues of its Hessian at its critical points. What is the corresponding concept of (non-)degeneracy of critical points when $k\geq 2$? Is there a normal (quadratic) form of $f$ near its nondengerate critical points when $k\geq 2$, as given in the classical Morse lemma? Any references will also be appreciated.

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    $\begingroup$ I think you might want Cerf theory. $\endgroup$ – Ben McKay May 31 at 14:28
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There is a vast literature on singularities, like:

  • Gibson, Christopher G.; Wirthmüller, Klaus; du Plessis, Andrew A.; Looijenga, Eduard J. N. Topological stability of smooth mappings. Lecture Notes in Mathematics, Vol. 552. Springer-Verlag, Berlin-New York, 1976

  • Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Volume 1. Classification of critical points, caustics and wave fronts. Birkhäuser 1985, 2012

  • Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Volume 2. Monodromy and asymptotics of integrals. Birkhäuser 1988, 2012.

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The answer in

Modification of Morse lemma with two functions

shows that this doesn't work in general (depending on what you mean)

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