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2011
This chapter studies conformal curvature tensors of a pseudo-Riemannian metric g. These are defined in terms of the covariant derivatives of the curvature tensor of an ambient metric in normal form relative to g. Their transformation laws under conformal change are given in terms of the action of a subgroup of the conformal group O(p + 1, q + 1) on ...
Charles Fefferman, C. Robin Graham
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This chapter studies conformal curvature tensors of a pseudo-Riemannian metric g. These are defined in terms of the covariant derivatives of the curvature tensor of an ambient metric in normal form relative to g. Their transformation laws under conformal change are given in terms of the action of a subgroup of the conformal group O(p + 1, q + 1) on ...
Charles Fefferman, C. Robin Graham
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On Classification of Curvature Tensor
American Journal of Physics, 1967A new method for classification of the gravitational field is presented. It is shown that this scheme leads to exactly the same types of gravitational fields as previously obtained by Petrov, Penrose, and Sachs, etc. Some physical applications are also given.
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2013
Suppose we have a coordinate system \({x}^{\mu }\) in a region of an \(n\)-dimensional Riemann (or pseudo-Riemann) manifold [20]. Components of the metric tensor \(g_{\mu \nu }\) are given as functions of \({x}^{\mu }\). We want to calculate the Riemann curvature tensor \({R}^{\mu }\,_{\nu \alpha \beta }\) and related quantities (the Ricci tensor \(R_{\
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Suppose we have a coordinate system \({x}^{\mu }\) in a region of an \(n\)-dimensional Riemann (or pseudo-Riemann) manifold [20]. Components of the metric tensor \(g_{\mu \nu }\) are given as functions of \({x}^{\mu }\). We want to calculate the Riemann curvature tensor \({R}^{\mu }\,_{\nu \alpha \beta }\) and related quantities (the Ricci tensor \(R_{\
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Curvature Tensors on Complex Lagrange Spaces
2003Let \((M,L)\) be a complex Lagrange space and let \(T_\mathbb{C} M\) be the complexification of the tangent bundle \(TM\), which is decomposed at each point of \(M\) into \(T_\mathbb{C} M=T'M+T''M\). The author expresses the local coefficients, in the adapted frame of the complex Chern-Lagrange nonlinear connection, of the Levi-Civita connection ...
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Some Remarks on Tensor Fields and Curvature
The Annals of Mathematics, 1952S. Bochner [1, 2, 3] has recently developed a beautiful theory on curvature and Betti numbers of an orientable compact Riemannian space V. with positive definite metric. He starts from the lemma: In V, , if the Laplacian Azp= gt'j;j; j _ 0 everywhere for a certain scalar (p, then we have Asp = 0 everywhere.
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Relativity, Tensors, and Curvature
2011Heuristics of Einstein's Theory What does g00 have to do with gravitation? The Metric Potentials Einstein's general theory of relativity is primarily a replacement for Newtonian gravitation and a generalization of special relativity. It cannot be “derived”; we can only speculate, with Einstein, by heuristic reasoning, how such a generalization ...
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Concircular Curvature Tensor and Fluid Spacetimes
International Journal of Theoretical Physics, 2009Zafar Ahsan +2 more
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On an Almost C(α)-Manifold Satisfying Certain Conditions on the Concircular Curvature Tensor
Pure and Applied Mathematics Journal, 2015Mehmet Atc̣Eken, Umit Yildirim
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