Topic
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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TL;DR: In this article, 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 $\ensuremath{-}1\mathrm{PN}$ 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.
145 citations
TL;DR: In this article, the Euler-Poincare equation associated with the geodesic flow of the right invariant metric on D sμ, the group of volume preserving Hilbert diffeomorphisms of classHs, was generalized to the setting of ann-dimensional compact Riemannian manifold.
145 citations
TL;DR: In this article, it was shown that the Ricci flow converges to a metric with constant bisectional curvature if and only if the curvature of the initial metric is positive.
Abstract: In this paper, we prove that if M is a Kahler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and the curvature is positive somewhere, then the Kahler-Ricci flow converges to a Kahler-Einstein metric with constant bisectional curvature. In a subsequent paper [7], we prove the same result for general Kahler-Einstein manifolds in all dimension. This gives an affirmative answer to a long standing problem in Kahler Ricci flow: On a compact Kahler-Einstein manifold, does the Kahler-Ricci flow converge to a Kahler-Einstein metric if the initial metric has a positive bisectional curvature? Our main method is to find a set of new functionals which are essentially decreasing under the Kahler Ricci flow while they have uniform lower bounds. This property gives the crucial estimate we need to tackle this problem.
145 citations
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TL;DR: In this paper, the compatibility of two definitions of generalized curvature bounds, the lower bound for sectional curvature in the sense of Alexandrov, and the upper bound for Ricci curvature for Lott-Villani-Sturm, was shown.
Abstract: Here I show the compatibility of two definitions of generalized curvature bounds: the lower bound for sectional curvature in the sense of Alexandrov and the lower bound for Ricci curvature in the sense of Lott-Villani-Sturm.
144 citations
TL;DR: In this article, the authors studied the properties of harmonic maps from a compact manifold M into a manifold N with non-positive curvature and showed that the space is a compact connected totally geodesic submanifold of N and is a point if r,(N) has no non-trivial abelian subgroup and the image of M is not a point or a circle.
142 citations