The constant rank theorem says that if $f\colon M→N$ is a smooth map whose rank equals some fixed $k≥0$ at any point of $M$, then, locally with respect to $M$ and $N$, the map $f$ assumes the easiest possible form: $f(x_1,…,x_k,…,x_m)=(x_1,…,x_k,0,…,0)$.

I am interested in a more general situation: the rank of $f$ is at most $k$ at any point of $M$ and can actually drop below $k$ at some points. The classification of local forms for such maps seems to be too difficult. However, in my case one can replace the map f with any map homotopic to it via a homotopy $M×[0,1]→N$ that is itself a smooth map of rank at most $k$. Thus, one can inquire about the possible local forms of smooth maps of rank at most $k$ up to smooth homotopy of rank at most $k$. I am mostly interested in the case $k < \operatorname{dim} M$.

Such a problem could presumably be solved using the techniques of catastrophe theory, but my knowledge of this area is virtually nonexistent.

**Is it possible to write down a finite list of local forms of smooth maps
of rank at most $k$ considered up to a smooth homotopy of rank at most $k$?**

This is motivated by the following question, which arises in connection to holonomy and parallel transport: given some $k<m$, can any smooth map $S^m→\mathbb{R}^n$ of rank at most $k$ be extended to a smooth map $D^{m+1}→\mathbb{R}^n$ of rank at most $k$?

stablemaps, which is exaclty the subject of singularity theory. $\endgroup$