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By Morse lemma for any $C^{\infty}$ function $f$ on $\mathbb R^2$ with Taylor series $(0,0)$ starting with $x^2+y^2$ one can find local $C^{\infty}$ coordinates $(x',y')$ such that locally $f(x',y')=...

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 ...