Manifolds

Abstract

Intuitively, a manifold is a generalization of curves and surfaces to higher dimensions. It is locally Euclidean in that every point has a neighborhood, called a chart, homeomorphic to an open subset of $\mathbb{R}^{\it{n}}$. The coordinates on a chart allow one to carry out computations as though in a Euclidean space, so that many concepts from $\mathbb{R}^{\it{n}}$, such as differentiability, point-derivations, tangent spaces, and differential forms, carry over to a manifold.