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Let $f:M\to N$ be a smooth surjective map between two compact manifolds. Is there always an open set $U\subset M$, s.t. on $U$, $f$ is submersive?

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By Sard's theorem there exists a regular value. Pick a point in its preimage and use the fact that having full rank is an open property to conclude that you find a open set where the differential has full rank.

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