[Math] A smooth map which has full rank at a point is locally a submersion/immersion.

continuitydifferential-geometry

Let $F:M\rightarrow N$
be a smooth map between manifolds and let $p\in M$
.

  1. If $dF_{p}$
    is surjective, then there exists a neighbourhood $U$
    of p
    such that $F|_{U}$
    is a submersion.

  2. If $dF_{p}$
    is injective, then there exists a neighbourhood $U$
    of $p$
    such that $F|_{U}$
    is an immersion.

I saw this theorem in Lee's Introduction to Smooth Manifolds. This is his proof:

enter image description here

What I don't understand is the part where says "by continuity". Is he talking about a function that is continuous? If so, what function is it?

Best Answer

In coordinates, $\Bbb d F$ is represented by a matrix. Since $\Bbb d F _p$ has full rank, there exists a minor in it of maximal dimension that is non-zero at $p$. The determinant of this minor is a continuous (in fact polynomial, being a sum of products of entries in the matrix) function of the coordinates, so if it's non-zero at $p$, by continuity it will be non-zero on some small neighbourhood of $p$, so the rank will be maximal on this neighbourhood.

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