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Suppose $F:M\to N$ is a smooth map, with $M,N$ smooth manifolds and $M$ connected. I want to show that if for each $p$ in $M$ there exists smooth charts near $p$ and $F(p)$ in which the coordinate representation of $F$ is linear, then $F$ has constant rank. The first step is to show that the rank of $F$ is constant in a neighborhood of each point because $F$ is linear. The next is to show $F$ has constant rank in all $M$ by connectedness hypothesis.

How can I use the connectedness of $M$ to say that the rank is constant on all of $M$? This comes from 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 $r$?2012-12-06
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    Note that the rank is constant locally.2012-12-06
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    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.2015-01-03

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