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

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 structure. 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