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Riemann curvature tensor

About: Riemann curvature tensor is a research topic. Over the lifetime, 6248 publications have been published within this topic receiving 138871 citations. The topic is also known as: Riemann–Christoffel tensor.


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Journal ArticleDOI
TL;DR: In this paper, it was shown that complete Riemannian manifolds with non-negative Ricci curvatures of dimension greater than or equal to three in which some Caffarelli-Kohn-Nirenberg type inequalities are satisfied are close to the Euclidean space.
Abstract: In this paper, we prove that complete open Riemannian manifolds with non-negative Ricci curvature of dimension greater than or equal to three in which some Caffarelli– Kohn–Nirenberg type inequalities are satisfied are close to the Euclidean space.

62 citations

Posted Content
TL;DR: The Einstein-Cartan Theory of gravity as mentioned in this paper is a modification of General Relativity Theory (GRT), allowing space-time to have torsion, in addition to curvature, and relating torsions to the density of intrinsic angular momentum.
Abstract: The Einstein--Cartan Theory (ECT) of gravity is a modification of General Relativity Theory (GRT), allowing space-time to have torsion, in addition to curvature, and relating torsion to the density of intrinsic angular momentum This modification was put forward in 1922 by Elie Cartan, before the discovery of spin Cartan was influenced by the work of the Cosserat brothers (1909), who considered besides an (asymmetric) force stress tensor also a moments stress tensor in a suitably generalized continuous medium

62 citations

15 Mar 2012
TL;DR: In this paper, a general class of quantum gravity-inspired, modified gravity theories are considered, where the Einstein-Hilbert action is extended through the addition of all terms quadratic in the curvature tensor coupled to scalar fields with standard kinetic energy.
Abstract: We consider a general class of quantum gravity-inspired, modified gravity theories, where the Einstein-Hilbert action is extended through the addition of all terms quadratic in the curvature tensor coupled to scalar fields with standard kinetic energy. This class of theories includes Einstein-Dilaton-Gauss-Bonnet and Chern-Simons modified gravity as special cases. We analytically derive and solve the coupled field equations in the post-Newtonian approximation, assuming a comparable-mass, spinning black hole binary source in a quasicircular, weak-field/slow-motion orbit. We find that a naive subtraction of divergent piece associated with the point-particle approximation is ill-suited to represent compact objects in these theories. Instead, we model them by appropriate effective sources built so that known strong-field solutions are reproduced in the far-field limit. In doing so, we prove that black holes in Einstein-Dilaton-Gauss-Bonnet and Chern-Simons theory can have hair, while neutron stars have no scalar monopole charge, in diametrical opposition to results in scalar-tensor theories. We then employ techniques similar to the direct integration of the relaxed Einstein equations to obtain analytic expressions for the scalar field, metric perturbation, and the associated gravitational wave luminosity measured at infinity. We find that scalar field emission mainly dominates the energy flux budget, sourcing electric-type (even-parity) dipole scalar radiation and magnetic-type (odd-parity) quadrupole scalar radiation, correcting the General Relativistic prediction at relative −1PN and 2PN orders. Such modifications lead to corrections in the emitted gravitational waves that can be mapped to the parameterized post-Einsteinian framework. Such modifications could be strongly constrained with gravitational wave observations.

62 citations

Journal ArticleDOI
TL;DR: In this paper, the curvature tensor of a real hypersurface M in complex two-plane Grassmannians G 2 (C m + 2 ) was derived from the equation of Gauss.

62 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202364
2022152
2021169
2020163
2019174
2018180