# Morse theory for vector-valued functions

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.

• I think you might want Cerf theory. May 31, 2020 at 14:28

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.