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.

## 2 Answers

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.

The answer in

Modification of Morse lemma with two functions

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