Manifold (geometry)

A manifold is an abstract mathematical space that looks locally like Euclidean space, but globally may have a very different structure. An example of this is a sphere: if one is very close to the surface of the sphere, it looks like a flat plane, but globally the sphere and plane are very different. Other examples of manifolds include lines and circles, and more abstract spaces such as the orthogonal group $O(n)$

The concept of a manifold is very important within mathematics and physics, and is fundamental to certain fields such as differential geometry, Riemannian geometry and General Relativity.

The most basic manifold is a topological manifold, but additional structures can be defined on the manifold to create objects such differentiable manifolds and Riemannian manifolds.

Mathematical Definition

In topology, a manifold of dimension $n$, or an n-manifold, is defined as a Hausdorff space where an open neighbourhood of each point is homeomorphic to $\scriptstyle \mathbb {R} ^{n}$.