I have found the following description on some notes of mathematical physics:
I have the classical equation for constraints $\Phi(x)=0$ for $x \in \mathbb{R}^d$, with $\Phi \in C^2(\mathbb{R}^d,\mathbb{R}^{d-m})$ with $rgD\Phi$ is max in every point. Then these notes continues like that: "...the kernel of $\Phi$ has to be $m$-dimensional..."
I know that the question seems to be too much general, but what is the kernel of a general application like the one above? I mean, in this paper there are not more information about $\Phi$, what I have written is all we know about $\Phi$ and I didn't find anything about Ker of applications which are not linear or preserve "rigid" structure.