An n-dimensional manifold (or n-manifold) M is a topological space such that every point in M has a neighbourhood that is homeomorphic to . These homeomorphisms induce a coordinatization of M, and it is further required that the coordinatization is continuous.
An alternate definition constructs the manifolds over the complex numbers instead of the real numbers. An n-dimensional complex manifold N is a topological space such that every point in N has a neighbourhood that is homeomorphic to Cn and whose coordinatization by these homeomorphisms is holomorphic (analytic).
Constructing new manifolds from old
- Suppose that is a differentiable function. Then f − 1(y) is a smooth manifold if y is a regular value of f.