Manifolds

DEFINITION

• General:

1. Simply an amorphous collection of points generally.
2. Any set that can be continuously parametrized. Parametrized: number of independent coordinate manifolds (dimension of Manifolds).

• In physics: Differential Manifolds

1. Continuous: neighborhood differ infinitesimally.
2. Differentiable: possible to define scalar field at each point of the manifolds that can be differentiated everywhere.

• General Relativity: Riemannian Manifolds

: this is an intrinsic property of manifolds (M).

That is called Riemannian if only ;

If , is called Pseudo-Riemannian Manifold.

Tangent Space of this Manifold is called Pseudo-Euclidean, for example Minkowski Space-time in Special Relativity.

PROPERTIES

The properties of manifolds based on Vector Space's point of view:

1. The neighborhood of some specified point P on Manifold M, the line element takes the Euclidean form.
2. There are N-Dimensionally Euclidean space at point P locally in Riemannian M.

This easily can be visual read by Tangent Space.