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- Ricci_curvature abstract "In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space. The Ricci tensor is defined on any pseudo-Riemannian manifold, as a trace of the Riemann curvature tensor. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold (Besse 1987, p. 43).In relativity theory, the Ricci tensor is the part of the curvature of space-time that determines the degree to which matter will tend to converge or diverge in time (via the Raychaudhuri equation). It is related to the matter content of the universe by means of the Einstein field equation. In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. comparison theorem) with the geometry of a constant curvature space form. If the Ricci tensor satisfies the vacuum Einstein equation, then the manifold is an Einstein manifold, which have been extensively studied (cf. Besse 1987). In this connection, the Ricci flow equation governs the evolution of a given metric to an Einstein metric, the precise manner in which this occurs ultimately leads to the solution of the Poincaré conjecture.".
- Ricci_curvature wikiPageExternalLink 0606774.
- Ricci_curvature wikiPageExternalLink 0612107.
- Ricci_curvature wikiPageID "276582".
- Ricci_curvature wikiPageRevisionID "601054591".
- Ricci_curvature author "L.A. Sidorov".
- Ricci_curvature hasPhotoCollection Ricci_curvature.
- Ricci_curvature id "R/r081780".
- Ricci_curvature id "r/r081800".
- Ricci_curvature title "Ricci curvature".
- Ricci_curvature title "Ricci tensor".
- Ricci_curvature subject Category:Curvature_(mathematics).
- Ricci_curvature subject Category:Riemannian_geometry.
- Ricci_curvature subject Category:Tensors_in_general_relativity.
- Ricci_curvature type Abstraction100002137.
- Ricci_curvature type Cognition100023271.
- Ricci_curvature type Concept105835747.
- Ricci_curvature type Content105809192.
- Ricci_curvature type Idea105833840.
- Ricci_curvature type PsychologicalFeature100023100.
- Ricci_curvature type Quantity105855125.
- Ricci_curvature type Tensor105864481.
- Ricci_curvature type TensorsInGeneralRelativity.
- Ricci_curvature type Variable105857459.
- Ricci_curvature comment "In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space.".
- Ricci_curvature label "Ricci curvature".
- Ricci_curvature label "Ricci-tensor".
- Ricci_curvature label "Tenseur de Ricci".
- Ricci_curvature label "Tensor de Ricci".
- Ricci_curvature label "Tensor de curvatura de Ricci".
- Ricci_curvature label "Tensore di curvatura di Ricci".
- Ricci_curvature label "Тензор Риччи".
- Ricci_curvature label "リッチテンソル".
- Ricci_curvature label "里奇曲率張量".
- Ricci_curvature sameAs Tensor_de_Ricci.
- Ricci_curvature sameAs Tenseur_de_Ricci.
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- Ricci_curvature sameAs リッチテンソル.
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- Ricci_curvature sameAs Ricci-tensor.
- Ricci_curvature sameAs Tensor_de_curvatura_de_Ricci.
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- Ricci_curvature sameAs Q1195879.
- Ricci_curvature sameAs Q1195879.
- Ricci_curvature sameAs Ricci_curvature.
- Ricci_curvature wasDerivedFrom Ricci_curvature?oldid=601054591.
- Ricci_curvature isPrimaryTopicOf Ricci_curvature.