Rank of a smooth map

Let $f:M\to N$ be a smooth map and let $p\in M$ . The rank of $f$ at $p$ is the rank of its differential

df_p:T_pM\longrightarrow T_{f(p)}N; \qquad \operatorname{rank}_p(f)=\dim(\operatorname{im}df_p).

Thus $0\leq \operatorname{rank}_p(f)\leq\min(\dim M,\dim N)$ . The map $f$ has constant rank $r$ on a subset if $\operatorname{rank}_p(f)=r$ at every point of that subset. The definition is independent of coordinates because changing charts composes the coordinate Jacobian with invertible linear maps.

Full-rank cases

If $\operatorname{rank}_p(f)=\dim M$ , then $df_p$ is injective; $f$ is an immersion at $p$ . If $\operatorname{rank}_p(f)=\dim N$ , then $df_p$ is surjective; $f$ is a submersion at $p$ . When the dimensions agree, either condition says that $df_p$ is an isomorphism, and the inverse function theorem makes $f$ a local diffeomorphism near $p$ .

Constant-rank normal form

If the rank is constantly $r$ on a neighborhood of $p$ , the constant-rank theorem gives local coordinates centered at $p$ and $f(p)$ in which

(x^1,\ldots,x^m)\longmapsto(x^1,\ldots,x^r,0,\ldots,0).

This normal form explains both the image dimension $r$ and the local fiber dimension $m-r$ . It is a theorem requiring local constancy of rank, not an alternative definition of the rank at a single point.

Semicontinuity and conventions

The rank function $p\mapsto\operatorname{rank}_p(f)$ is lower semicontinuous: the set where the rank is at least $r$ is open, since some $r\times r$ Jacobian minor remains nonzero nearby. Consequently the maximal-rank locus is open, but lower-rank loci may be singular. The rank–nullity theorem gives $\dim\ker df_p=\dim M-\operatorname{rank}_p(f)$ . “Rank of $f$ ” without a point should be used only when the rank is constant or when a stated convention means the maximum rank.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics 218, Springer, 2012. Publisher record. Relevant: Chapter 4, rank, immersion, submersion, and constant-rank theorems.
Morris W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, 1976. Publisher record. Relevant: Chapter 1, rank and local normal forms.