Manifold

A manifold is a topological space that locally resembles Euclidean space near each point.

As defined in Physics

An N-dimensional manifold of points is one for which N independent real coordinates $(x^{1}, x^{2},…,x^{N})$ are required to specify a point completely. These N coordinates are denoted collectively by $x^{a}$, where it is understood that $a\,=\,1,2,…,N$.

As an example, in $\mathbb{R}^{2}$ we have a 2-dimensional manifold of points descrived by the real coordinates $(x^{1}, x^{2})$.