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 generalization of the p-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor is introduced, called the (p, q)-curvatures.
Abstract: We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the (p, q)-curvatures. They are a generalization of the p-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for p = 0, the (0, q)-curvatures coincide with the H. Weyl curvature invariants, for p = 1 the (1, q)-curvatures are the curvatures of generalized Einstein tensors, and for q = 1 the (p, 1)-curvatures coincide with the p-curvatures. Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension n > 4, and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.
58 citations
TL;DR: The structures in the eigenvalue and eigenvector fields of the gradient tensor and how these structures can be used to infer the behaviors of the velocity field are described.
Abstract: The gradient of a velocity vector field is an asymmetric tensor field which can provide critical insight that is difficult to infer from traditional trajectory-based vector field visualization techniques. We describe the structures in the eigenvalue and eigenvector fields of the gradient tensor and how these structures can be used to infer the behaviors of the velocity field. To illustrate the structures in asymmetric tensor fields, we introduce the notions of eigenvalue and eigenvector manifolds. These concepts afford a number of theoretical results that clarify the connections between symmetric and antisymmetric components in tensor fields. In addition, these manifolds naturally lead to partitions of tensor fields, which we use to design effective visualization strategies. Both eigenvalue manifold and eigenvector manifold are supported by a tensor reparameterization with physical meaning. This allows us to relate our tensor analysis to physical quantities such as rotation, angular deformation, and dilation, which provide physical interpretation of our tensor-driven vector field analysis in the context of fluid mechanics. To demonstrate the utility of our approach, we have applied our visualization techniques and interpretation to the study of the Sullivan vortex as well as computational fluid dynamics simulation data.
58 citations
TL;DR: In this article, the Friedmann equations for the apparent horizon of FRW spacetime in f(R,T,RμνTμν) gravity were studied and the generalized second law of thermodynamics was established in a more comprehensive form.
Abstract: We study first and second laws of black hole thermodynamics at the apparent horizon of FRW spacetime in f(R,T,RμνTμν) gravity, where R, Rμν are the Ricci scalar and Riemann tensor and T is the trace of the energy-momentum tensor Tμν. We develop the Friedmann equations for any spatial curvature in this modified theory and show that these equations can be transformed to the form of Clausius relation ThSeff = δ. Here Th is the horizon temperature, Seff is the entropy which contains contributions both from horizon entropy and additional entropy term introduced due to the non-equilibrating description and δ is the energy flux across the horizon. The generalized second law of thermodynamics is also established in a more comprehensive form and one can recover the corresponding results in Einstein, f(R) and f(R,T) gravities. We discuss GSLT in the locality of assumption that temperature of matter inside the horizon is similar to that of horizon. Finally, we consider particular models in this theory and generate constraints on the coupling parameter for the validity of GSLT.
58 citations
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TL;DR: In this paper, generalized tensor analysis is employed to introduce a nonlinear distributional pseudo-Riemannian geometry, and the notion of geodesics of a generalized metric is defined.
Abstract: Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a ``Fundamental Lemma of (pseudo-)Riemannian geometry'' in this setting and define the notion of geodesics of a generalized metric. Finally, we present applications of the resulting theory to general relativity.
58 citations
TL;DR: In this article, the authors investigated the properties of the Ma-Trudinger-Wang nonlocal curvature tensor in the case of surfaces and proved that a strict form of the condition is stable under C4 perturbation if the nonfocal domains are uniformly convex.
Abstract: We investigate the properties of the Ma–Trudinger–Wang nonlocal curvature tensor in the case of surfaces. In particular, we prove that a strict form of the Ma–Trudinger– Wang condition is stable under C4 perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the Ma–Trudinger–Wang condition. As a corollary of our results, optimal transport maps on a “sufficiently flat” ellipsoid are in general nonsmooth.
58 citations