Dimension of a manifold ($\dim M$)

A topological manifold has dimension $n$ when its manifold charts have coordinate domains open in ${\mathbb{R}}^n$. For a smooth manifold, each tangent space has dimension $n$, and a tangent bundle has vector bundle rank $n$.

The codimension is the difference between the dimension of a manifold and that of its submanifold. For complex submanifolds, complex codimension one means real codimension two.