Search results for "Levi-Civita connection"

showing 3 items of 3 documents

The Riemannian manifold of all Riemannian metrics

1991

In this paper we study the geometry of (M, G) by using the ideas developed in [Michor, 1980]. With that differentiable structure on M it is possible to use variational principles and so we start in section 2 by computing geodesics as the curves in M minimizing the energy functional. From the geodesic equation, the covariant derivative of the Levi-Civita connection can be obtained, and that provides a direct method for computing the curvature of the manifold. Christoffel symbol and curvature turn out to be pointwise in M and so, although the mappings involved in the definition of the Ricci tensor and the scalar curvature have no trace, in our case we can define the concepts of ”Ricci like cu…

Mathematics - Differential GeometryChristoffel symbolsGeneral MathematicsPrescribed scalar curvature problem58D17 58B20Mathematical analysisCurvatureLevi-Civita connectionFunctional Analysis (math.FA)Mathematics - Functional Analysissymbols.namesakeDifferential Geometry (math.DG)symbolsFOS: MathematicsSectional curvatureMathematics::Differential GeometryExponential map (Riemannian geometry)Ricci curvatureScalar curvatureMathematics
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Some remarks on minimal surfaces in riemannian manifolds

1970

Pure mathematicsCurvature of Riemannian manifoldsRiemannian submersionApplied MathematicsGeneral Mathematics010102 general mathematicsMathematical analysisFundamental theorem of Riemannian geometryRiemannian geometry01 natural sciencesLevi-Civita connectionsymbols.namesakeRicci-flat manifold0103 physical sciencessymbolsMinimal volume010307 mathematical physicsSectional curvature0101 mathematicsMathematicsCommunications on Pure and Applied Mathematics
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Feuilletages Riemanniens singuliers

2006

Abstract We prove that a singular foliation on a compact manifold admitting an adapted Riemannian metric for which all leaves are minimal must be regular. To cite this article: V. Miquel, R.A. Wolak, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Pure mathematicsMathematical analysisGeneral MedicineRiemannian geometryFundamental theorem of Riemannian geometryPseudo-Riemannian manifoldLevi-Civita connectionsymbols.namesakesymbolsMinimal volumeMathematics::Differential GeometryExponential map (Riemannian geometry)Ricci curvatureScalar curvatureMathematicsComptes Rendus Mathematique
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