Smooth map

Smooth functions on and between manifolds

Preimages and images of smooth maps are, in general, not manifolds without additional assumptions. Preimages of regular values are manifolds; this means that, for a smooth map ${\displaystyle F:M\to N}$ and a value ${\displaystyle q\in N}$, the differential ${\displaystyle dF_{p}:T_{p}M\to T_{q}N}$ is surjective at every point ${\displaystyle p\in F^{-1}(q)}$. This is the preimage theorem. Similarly, the image of an embedding is an embedded submanifold.