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Holonomy Group
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features. Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy. The most common forms of holonomy are for connections possessing some kind of symmetry. Important examples include: holonomy of the Levi-Civita connection in Riemannian geometry (called Riemannian holonomy), holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles. In each of these cases, the holonomy of the connection can be identified with a Lie g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis. In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (''M'', ''g'') is said to be symmetric if and only if, for each point ''p'' of ''M'', there exists an isometry of ''M'' fixing ''p'' and acting on the tangent space T_pM as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
