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Let $f:N\to M$ be a smooth map between two manifolds of dimension $n,m$. If $f_*$, the differential of $f$ is injective at $p$ we say that $f$ is an immersion. I want to show that $f$ has constant rank $n$ in some neighborhood of $p$. I have no idea how to get started. Any help is appreciated. Thanks!

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  • $\begingroup$ If the differential is injective at all points, the rank is $n$ at all points. This is clear, no? $\endgroup$ Jun 29 '17 at 3:46
  • $\begingroup$ @Mariano: One is assuming only that the differential is injective at a single point. $\endgroup$ Jun 29 '17 at 14:03
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HINT: If a continuously varying $m\times n$ matrix has rank $n$ at some point $p$, show that it has rank $n$ nearby. (Remember that determinant is continuous.)

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  • $\begingroup$ Thank you for the hint. But why consider the determinant? $\endgroup$
    – Heisenberg
    Jun 29 '17 at 3:06
  • $\begingroup$ Well ... How do you determine when an $m\times n$ matrix has rank $n$? (And before you tell me something about echelon form, remember that you want a criterion that you can apply when you "wiggle" the matrix.) $\endgroup$ Jun 29 '17 at 3:08
  • $\begingroup$ Isn't it just the dimension of the column space? $\endgroup$
    – Heisenberg
    Jun 29 '17 at 3:11
  • $\begingroup$ Yes. Now how do you prove that that dimension stays constant when you move the matrix nearby? $\endgroup$ Jun 29 '17 at 3:13
  • $\begingroup$ I am still kind of lost. Do I have to show that the determinant of the Jacobian is non zero for a nxn submatrix of the Jacobian? $\endgroup$
    – Heisenberg
    Jun 29 '17 at 3:18

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