Let $f: M \to N$ be a smooth maps between smooth manifolds. Then $f$ is a submersion (by definition) if its differential is also surjective. Now suppose $f$ is surjective. Is it possible that the surjective map $f$ fails to be a submersion on a set in $N$ of measure non-zero? If so, what is such a map?

Suppose the manifolds $M$ and $N$ are non-compact. Does this change the previous answer?


  • 1
    $\begingroup$ Sard's theorem. $\endgroup$
    – BMann
    Mar 30, 2011 at 21:50

2 Answers 2


Sard's Theorem


If you relax the smoothness assumption, you can construct a surjection $f:\mathbb{R}^{n+1}\to\mathbb{R}^n$, $n\geq 2$ of class $C^1$ such that ${\rm rank}\, Df\leq 1$ everywhere, see:

R. Kaufman, A singular map of a cube onto a square. J. Differential Geom. 14 (1979), no. 4, 593–594 (1981). (MathSciNet review.)


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