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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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TL;DR: A geometrical framework for double field theory in which generalized Riemann and torsion tensors are defined without reference to a particular basis is introduced and it is shown that it contains the conventional Ricci tensor and scalar curvature but not the full Riem Mann tensor.
Abstract: We introduce a geometrical framework for double field theory in which generalized Riemann and torsion tensors are defined without reference to a particular basis. This invariant geometry provides a unifying framework for the frame-like and metric-like formulations developed before. We discuss the relation to generalized geometry and give an “index-free” proof of the algebraic Bianchi identity. Finally, we analyze to what extent the generalized Riemann tensor encodes the curvatures of Riemannian geometry. We show that it contains the conventional Ricci tensor and scalar curvature but not the full Riemann tensor, suggesting the possibility of a further extension of this framework.

105 citations

Journal ArticleDOI
TL;DR: In this article, the Ricci flat metrics with nonzero parallel spinors are shown to be stable in the direction of changes in conformal structures, which is a local version of the HMM03 result.
Abstract: Inspired by the recent work [HHM03], we prove two stability results for compact Riemannian manifolds with nonzero parallel spinors. Our first result says that Ricci flat metrics which also admit nonzero parallel spinors are stable (in the direction of changes in conformal structures) as the critical points of the total scalar curvature functional. Our second result, which is a local version of the first one, shows that any metric of positive scalar curvature cannot lie too close to a metric with nonzero parallel spinor. We also prove a rigidity result for special holonomy metrics. In the case of SU(m) holonomy, the rigidity result implies that scalar flat deformations of Calabi-Yau metric must be Calabi-Yau. Finally we explore the connection with a positive mass theorem of [D03], which presents another approach to proving these stability and rigidity results.

105 citations

Journal ArticleDOI
TL;DR: In this paper, the authors study the Horndeski vector-tensor theory that leads to second order equations of motion and contains a non-minimally coupled abelian gauge vector field.
Abstract: We study the Horndeski vector-tensor theory that leads to second order equations of motion and contains a non-minimally coupled abelian gauge vector field. This theory is remarkably simple and consists of only 2 terms for the vector field, namely: the standard Maxwell kinetic term and a coupling to the dual Riemann tensor. Furthermore, the vector sector respects the U(1) gauge symmetry and the theory contains only one free parameter, M 2 , that controls the strength of the non-minimal coupling. We explore the theory in a de Sitter spacetime and study the presence of instabilities and show that it corresponds to an attractor solution in the presence of the vector field. We also investigate the cosmological evolution and stability of perturbations in a general FLRW spacetime. We find that a sufficient condition for the absence of ghosts is M 2 > 0. Moreover, we study further constraints coming from imposing the absence of Laplacian instabilities. Finally, we study the stability of the theory in static and spherically symmetric backgrounds (in particular, Schwarzschild and Reissner-Nordstrom-de Sitter). We find that the theory, quite generally, do have ghosts or Laplacian instabilities in regions of spacetime where the non-minimal interaction dominates over the Maxwell term. We also calculate the propagation speed in these spacetimes and show that superluminality is a quite generic phenomenon in this theory. ©2013 IOP Publishing Ltd and Sissa Medialab srl.

104 citations

Journal ArticleDOI
TL;DR: The Dirichlet sub-solution for nonnegative sectional, Ricei, and bisectional curvature problems was studied in this paper, where the main point of this procedure is to sidestep arguments involving continuous functions by working with differentiable functions alone.
Abstract: A standard technique in classical analysis for the study of eontinous sub-solutions of the Dirichlet problem for second order operators may be illustrated as follows. Suppose it is to be shown that a continuous real function ](x) is convex (respectively, striely convex) at x0; then it suffices to produce a C ~ function g(x) such that g(x)<<.](x) near x 0 and g(Xo) =/(x0), and such that 9\"(xo) >/0 (respectively g\"(xo) >1 some fixed positive constant). The main point of this procedure is to sidestep arguments involving continuous functions by working with differentiable functions alone. Now in global differential geometry, the functions that naturally arise are often continuous but not differentiable. Since much of geometric analysis reduces to second order elliptic problems, this technique then recommends itself as a natural tool for overcoming this difficulty with the lack of differentiability. In a limited way, this technique has indeed appeared in several papers in complex geometry (e.g. Ahlfors [1], Takeuchi [20], Elenewajg [7] and Greene-Wu [11]; cf. also Suzuki [19]). The main purpose of this paper is to broaden and deepen the scope of this method by making it the central point of a general study of nonnegative sectional, Ricei or bisectional curvature. The following are the principal theorems; the relevant definitions can be found in Section 1. Let M be a noncompact complete Riemannian manifold and let 0 E M be fixed. Let {Ct}tG1 be a family of closed subsets of M indexed by a subset I of R. Assume that et = d(0, C t ) ~ as t ~ , where d(p, q) will always denote the distance between p, qEM relative to the Riemannian metric. The family of functions ~t: M-~R defined by ~t(P)=

104 citations


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