Rank (differential topology)

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In mathematics, the rank of a differentiable map f : M → N between differentiable manifolds at a point p ∈ M is the rank of the derivative of f at p. Recall that the derivative of f at p is a linear map

$Df_p : T_p M \to T_{f(p)}N\,$

from the tangent space at p to the tangent space at f(p). As a linear map between vector spaces it has a well-defined rank, which is just the dimension of the image in T_f(p)N:

$\operatorname{rank}(f)_p = \dim(\operatorname{im}(Df_p)).$

[edit] Constant rank maps

A differentiable map f : M → N is said to have constant rank if the rank of f is the same for all p in M. Constant rank maps have a number of nice properties and are an important concept in differential topology.

Three special cases of constant rank maps occur. A constant rank map f : M → N is

an immersion if rank f = dim M (i.e. the derivative is everywhere injective),
a submersion if rank f = dim N (i.e. the derivative is everywhere surjective),
a local diffeomorphism if rank f = dim M = dim N (i.e. the derivative is everywhere bijective).

The map f itself need not be injective, surjective, or bijective for these conditions to hold, only the behavior of the derivative is important. For example, there are injective maps which are not immersions and immersions which are not injections. However, if f : M → N is a smooth map of constant rank then

if f is injective it is an immersion,
if f is surjective it is a submersion,
if f is bijective it is a diffeomorphism.

Constant rank maps have a nice description in terms of local coordinates. Suppose M and N are smooth manifolds of dimensions m and n respectively, and f : M → N is a smooth map with constant rank k. Then for all p in M there exist coordinates (x¹, ..., x^m) centered at p and coordinates (y¹, ..., yⁿ) centered at f(p) such that f is given by