constant rank theorem

Statement

Let and be smooth manifolds and a smooth map with constant rank . Then the image is an embedded submanifold. Furthermore, the tangent space is the image of the differential .

Another (easier) statement

Let and be smooth manifolds, and a smooth map with constant rank.

1. If is surjective, then it is a (smooth) submersion.
2. If is injective, then it is a (smooth) immersion.
3. If is bijective, then it is a diffeomorphism.