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# Topic: Riemannian manifold

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 Riemannian manifold   (Site not responding. Last check: 2007-09-10) In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. Usually a Riemannian manifold is defined as a smooth manifold with a smooth section of positive-definite quadratic forms on the tangent bundle. With this definition of length, every connected Riemannian manifold M becomes a metric space (and even a length metric space) in a natural fashion: the distance d(x, y) between the points x and y of M is defined as publicliterature.org /en/wikipedia/r/ri/riemannian_manifold.html   (523 words)

 Manifold In physics, differentiable manifolds serve as the phase space in classical mechanics and four dimensional pseudo-Riemannian manifolds are used to model spacetime in general relativity. Requiring a manifold to be Hausdorff may seem strange; it is tempting to think that being locally homeomorphic to a Euclidean space implies being a Hausdorff space. A Riemannian manifold is a differentiable manifold on which the tangent spaces are equipped with inner products in a differentiable fashion. www.fact-index.com /m/ma/manifold_1.html   (1103 words)

 Manifold   (Site not responding. Last check: 2007-09-10) A pseudo-Riemannian manifold is a variant of Riemannian manifold where the metric tensor is allowed to have an indefinite signature (as opposed to a positive-definite one). A symplectic manifold is a Manifold equipped with a closed, nondegenerate, alternating 2-form. A Kähler manifold is a Manifold which simultaneously carries a Riemannian structure, a symplectic structure, and a complex structure which are all compatible in some suitable sense. manifold.iqnaut.net   (2005 words)

 physics - Pseudo-Riemannian manifold   (Site not responding. Last check: 2007-09-10) In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, (0,2) tensor which is nondegenerate at each point on the manifold. The key difference between a Riemannian metric and a pseudo-Riemannian metric is that a pseudo-Riemannian metric need not be positive-definite, merely nondegenerate. The signature of a pseudo-Riemannian manifold is just the signature of the metric (one should insist that the signature is the same on every connected component). www.physicsdaily.com /physics/Pseudo-Riemannian_manifold   (371 words)

 manifold Every manifold has a dimension, which is the number of coordinates needed to specify it in the local coordinate system. Differentiable manifolds are used in mathematics to describe geometrical objects, and are also the most natural and general settings in which to study differentiability. In physics, differentiable manifolds serve as the phase space in classical mechanics, while four dimensional pseudo-Riemannian manifolds are used to model spacetime in general relativity. www.daviddarling.info /encyclopedia/M/manifold.html   (286 words)

 Riemannian manifold - Wikipedia, the free encyclopedia In Riemannian geometry, a Riemannian manifold (M,g) (with Riemannian metric g) is a real differentiable manifold M in which each tangent space is equipped with an inner product g in a manner which varies smoothly from point to point. The tangent bundle of a smooth manifold M (or indeed, any vector bundle over a manifold) is, at a fixed point, just a vector space and each such space can carry an inner product. In Riemannian manifolds, the notions of geodesic completeness, topological completeness and metric completeness are the same: that each implies the other is the content of the Hopf-Rinow theorem. en.wikipedia.org /wiki/Riemannian_manifold   (529 words)

 Riemannian geometry - Wikipedia, the free encyclopedia In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i.e. Gauss–Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ(M) where χ(M) denotes the Euler characteristic of M. The set of all Riemannian manifolds with positive Ricci curvature and diameter at most D is pre-compact in the Gromov-Hausdorff metric. en.wikipedia.org /wiki/Riemannian_geometry   (845 words)

 Pseudo Riemannian manifold - KnowledgeIsFun.com In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. Just as Riemannian manifolds may be thought of as being locally modeled on Euclidean space, Lorentzian manifolds are locally modeled on Minkowski space. A pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric www.knowledgeisfun.com /P/Ps/Pseudo-Riemannian-manifold.php   (476 words)

 PlanetMath: Riemannian manifold Indeed, it is possible to define a Riemannian structure on a manifold Cross-references: distance, rectifiable curves, infimum, distance metric, coordinate chart, atlas, manifold, riemannian structure, matrix, fix, smooth functions, components, vector fields, frame, 1-forms, coframe, open subset, local coordinates, function, global sections, sheaf, cotangent bundle, positive definite, symmetric, bilinear form, point, field, type, tensor This is version 14 of Riemannian manifold, born on 2002-09-12, modified 2006-10-22. planetmath.org /encyclopedia/RiemannianMetric.html   (237 words)

 Pseudo-Riemannian manifold - Wikipedia, the free encyclopedia Lorentzian manifolds occur in the general theory of relativity as models of curved 4-dimensional spacetime. A pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric (0,2) tensor which is nondegenerate at each point on the manifold. The signature of a pseudo-Riemannian manifold is just the signature of the metric on any given tangent space (one should insist that the signature is the same on every connected component). en.wikipedia.org /wiki/Pseudo-Riemannian_manifold   (447 words)

 Riemannian Geometry An N-dimensional Riemannian manifold is characterized by a second-order metric tensor g If there exists a coordinate system at a point on the manifold such that the metric components are constant in the first and second order, then the manifold is said to be totally flat at that point (not just asymptotically flat). The “connection” of this manifold is customarily expressed in the form of Christoffel symbols. www.mathpages.com /rr/s5-07/5-07.htm   (2194 words)

 Riemannian Manifold The integrated dimension reveals the manifest world as a Riemannian multidimensional manifold, and shows the hierarchical relationships of the functioning of its various dimensions. Viewing our consciousness in terms of its dimensions, we see that it is not a three-or four-dimensional field, but a multidimensional manifold, a manifold in the sense that it is a dynamic structure of dynamic structures. This manifold is characterized by a nonlinear -- Riemannian -- geometry, in that all the dimensions open up to all the others in nonlinear ways. www.ahalmaas.com /glossary/r/riemannian.htm   (309 words)

 Citations: the curvatura integra in a Riemannian manifold - Chern (ResearchIndex)   (Site not responding. Last check: 2007-09-10) Citations: the curvatura integra in a Riemannian manifold - Chern (ResearchIndex) S.S. Chern, On the curvatura integra in a Riemannian manifold, Annals of Math., 46, (1945), 674-684. Here i are the eigenvalues of the 2 nd fundamental form A and the indices i run over an orthonormal basis of the tangent spaces to D: The sign on A is chosen so that i 0 for convex domains; K denotes sectional curvature. citeseer.ist.psu.edu /context/1881000/0   (206 words)

 Amazon.com: The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds (London Mathematical ...   (Site not responding. Last check: 2007-09-10) Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics) by John M. Lee This text on analysis on Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. The text is aimed at students who have had a first course in differentiable manifolds, and the author develops the Riemannian geometry used from the beginning. www.amazon.com /Laplacian-Riemannian-Manifold-Introduction-Mathematical/dp/0521468310   (1139 words)

 IngentaConnect Heat Content of a Riemannian Manifold with a Perfect Conducting B...   (Site not responding. Last check: 2007-09-10) IngentaConnect Heat Content of a Riemannian Manifold with a Perfect Conducting B... Let M be a compact smooth Riemannian manifold with smooth boundary. We establish the existence of an asymptotic series for the heat content of M with a perfect conducting boundary and show that the coefficients in the series are non-local invariants which are recursively determined by the coefficients for the series with corresponding zero Dirichlet boundary condition. www.ingentaconnect.com /content/klu/pota/2003/00000019/00000001/05095833   (153 words)

 Optimal Regularity Of Harmonic Maps From A Riemannian Manifold Into A Static Lorentzian Manifold (ResearchIndex)   (Site not responding. Last check: 2007-09-10) Optimal Regularity Of Harmonic Maps From A Riemannian Manifold Into A Static Lorentzian Manifold (1997) Abstract: this paper, we give an optimal regularity result for some class of weakly harmonic maps from a Riemannian manifold (Update) 0.2: The Dirac operator on Lorentzian spin manifolds and the Huygens.. citeseer.ist.psu.edu /65294.html   (331 words)

 Springer Online Reference Works » Encyclopaedia of Mathematics » R » Riemannian manifold A differentiable manifold provided with a Riemannian metric. Essentially, a Riemannian manifold is the same as a Riemannian space. eom.springer.de /r/r082170.htm   (47 words)

 CiteULike: ansobol's riemannian-manifold   (Site not responding. Last check: 2007-09-10) Local monotonicity and mean value formulas for evolving Riemannian manifolds Inf-convolution and regularization of convex functions on Riemannian manifolds of nonpositive curvature Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds www.citeulike.org /user/ansobol/tag/riemannian-manifold   (210 words)

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