Constant rank theorem - Imported Knowlpedia Content

Statement

Let $f:M^m\to N^n$ be a smooth map, and suppose its rank is constantly $r$ on a neighborhood of $p\in M$ . The constant rank theorem states that there are smooth coordinate charts $\varphi$ about $p$ and $\psi$ about $f(p)$ , both centered at the origin, such that

(\psi\circ f\circ\varphi^{-1})(x^1,\ldots,x^m) =(x^1,\ldots,x^r,0,\ldots,0).

Thus, locally and up to diffeomorphisms of source and target, $f$ is the projection onto $r$ coordinates followed by the inclusion into $n$ coordinates. Local constancy of rank near $p$ , not merely rank $r$ at $p$ , is the essential hypothesis.

Geometric consequences

After shrinking the charts, each nonempty local fiber of $f$ is an embedded submanifold of dimension $m-r$ , described by fixing the first $r$ coordinates. The local image is an embedded $r$ -dimensional submanifold of $N$ . Moreover,

T_x(f^{-1}(f(x)))=\ker df_x

throughout the constant-rank neighborhood. These conclusions are immediate from the displayed normal form.

Full-rank special cases

When $r=m$ , the normal form is a coordinate inclusion, giving the local model for a smooth immersion. When $r=n$ , it is a coordinate projection, giving the local model for a smooth submersion. If $m=n=r$ , it reduces to the inverse function theorem and $f$ is a local diffeomorphism.

Conventions and scope

Some texts call this result simply the “rank theorem.” A related local form remains available when the rank is only known to be $r$ at $p$ , provided a suitable $r\times r$ minor stays nonzero and additional target components are retained; the clean zero-component form above requires constant rank. See Lee, Theorem 4.12.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics 218, Springer, 2012. DOI record. Relevant: Chapter 4, the rank theorem and its immersion and submersion corollaries.
Victor Guillemin and Alan Pollack, Differential Topology, AMS Chelsea Publishing, 2010 reprint. DOI record. Relevant: Chapter 1, local forms of smooth maps.