# An immersion has constant rank in some neighborhood

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!

• If the differential is injective at all points, the rank is $n$ at all points. This is clear, no? Jun 29 '17 at 3:46
• @Mariano: One is assuming only that the differential is injective at a single point. Jun 29 '17 at 14:03

## 1 Answer

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.)

• Thank you for the hint. But why consider the determinant? Jun 29 '17 at 3:06
• 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.) Jun 29 '17 at 3:08
• Isn't it just the dimension of the column space? Jun 29 '17 at 3:11
• Yes. Now how do you prove that that dimension stays constant when you move the matrix nearby? Jun 29 '17 at 3:13
• 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? Jun 29 '17 at 3:18