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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 42. Chapters: Curvature, Riemann curvature tensor, Holonomy, Frenet-Serret formulas, Darboux frame, Torsion tensor, Ricci curvature, Radius of curvature, Gauss-Codazzi equations, Curvature of Riemannian manifolds, Gaussian curvature, Second fundamental form, Sectional curvature, Calculus of moving surfaces, Principal curvature, Scalar curvature, Menger curvature, Mean curvature, Curvature of a measure, Levi-Civita parallelogramoid, Torsion of a curve, Total curvature, Geodesic curvature, Curvature form, Non-positive curvature, Cocurvature. Excerpt: 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 group, the holonomy group. The holonomy of a connection is closely related to the curvature of the connection, via the Ambrose-Singer theorem. The study of Riemannian holonomy has led to a number of important developments. The holonomy was introduced by Cartan (1926) in order to study and classify symmetric spaces. It...
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