topics in manifold theory

A manifold is a space that is locally like ${ℝ}^n$, however lacking a preferred system of coordinates. Furthermore, a manifold can have global topological properties, such as non-contractible loops (http://planetmath.org/Curve), that distinguish it from the topologically trivial ${ℝ}^n$.

By imposing different restrictions on the transition functions of a manifold, one obtain different types of manifolds:

• topological manifolds

• $C^k$ manifolds, smooth manifolds

• real analytic manifold

• complex analytic manifold

• symplectic manifolds, where transition functions are symplectomorphisms. On such manifolds, one can formulate the Hamilton equations.

Special types of manifolds

• orientable manifolds

• manifolds with boundary

• compact manifolds

On manifolds, one can introduce more . Some examples are:

• Riemannian manifolds

• contact manifolds

• CR manifolds

• fiber bundles and sheaves

Examples

• space-time manifold in general relativity

• phase space in mechanics

• de Rham cohomology in algebraic topology

See also

For the formal definition click here (http://planetmath.org/Manifold)
http://en.wikipedia.org/wiki/ManifoldManifold entry at Wikipedia

Title	topics in manifold theory
Canonical name	TopicsInManifoldTheory
Date of creation	2013-03-22 14:11:04
Last modified on	2013-03-22 14:11:04
Owner	evin290 (5830)
Last modified by	evin290 (5830)
Numerical id	15
Author	evin290 (5830)
Entry type	Topic
Classification	msc 53-00