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:

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

Special types of manifolds

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orientable manifolds
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manifolds with boundary
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compact manifolds

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

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Riemannian manifolds
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contact manifolds
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CR manifolds
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fiber bundles and sheaves

Examples

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space-time manifold in general relativity
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phase space in mechanics
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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
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topics in manifold theory

Examples

See also