topics in manifold theory

A manifold is a space that islocally like $\\mathbb{R}^{n}$ , however lacking a preferred system ofcoordinates. Furthermore, a manifold can have global topologicalproperties, such as non-contractible loops (http://planetmath.org/Curve), that distinguish it fromthe topologically trivial $\\mathbb{R}^{n}$ .

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

•
topological manifolds
•
$C^{k}$ manifolds, smooth manifolds
•
real analytic manifold
•
complex analytic manifold
•
symplectic manifolds, where transition functionsare symplectomorphisms. On such manifolds, one can formulate theHamilton 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

单词	TopicsInManifoldTheory
释义	topics in manifold theory A manifold is a space that islocally like $\\mathbb{R}^{n}$ , however lacking a preferred system ofcoordinates. Furthermore, a manifold can have global topologicalproperties, such as non-contractible loops (http://planetmath.org/Curve), that distinguish it fromthe topologically trivial $\\mathbb{R}^{n}$ . By imposing different restrictions on the transition functions of a manifold, oneobtain different types of manifolds: • topological manifolds • $C^{k}$ manifolds, smooth manifolds • real analytic manifold • complex analytic manifold • symplectic manifolds, where transition functionsare symplectomorphisms. On such manifolds, one can formulate theHamilton 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
随便看	Church integer ChurchRosserProperty Church-Rosser property ChuSpace Chu space ChuTransform Chu transform Cinfty0UIsNotEmpty Circle circle CircleHasOneCenter UpperBoundOnvarthetan upper bound on ϑ⁢(n) UpperHalfPlane upper half plane UpperNilradical upper nilradical UpperSet upper set UpperSetOperationIsAClosureOperator upper set operation is a closure operator UpwardLowenheimSkolemTheorem upward Lowenheim-Skolem theorem UrysohnExtensionTheorem Urysohn extension theorem

topics in manifold theory

Examples

See also