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Suppose $F:M\to N$, $M,N$ smooth manifolds and $M$ connected. I proved that for each $p$ in $M$ there exists smooth charts near $p$ and $F(p)$ in which the coordinate rapresentation of $F$ is linear. This is sufficient to prove that the rank of $F$ is constant in a neighborhood of each point. How can I use the connectedness of $M$ to say that the rank is constant on all of $M$? This is a property on the book of "Introduction to Smooth Manifolds" by Lee.

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Fix an integer $r\leq\min(\dim M,\dim N)$. What can you say about the subset of points of $M$ where the rank of $F$ is $r$? – Olivier Bégassat Dec 6 '12 at 6:32
Note that the rank is constant locally. – lee Dec 6 '12 at 8:49
Hmm? Maybe I got your question wrong, but rank is NOT a local property. That a map is IMMERSION or SUBMERSION is. For example, consider $f(x) = x^3$, which has rank 1 everywhere but at zero. – chriseur Jan 3 '15 at 5:19

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