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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: In this article, the authors studied the geometric and topological consequences of the existence of a non-trivial Codazzi tensor on a given Riemannian manifold.
Abstract: for arbitrary vector fields X, Y, Z. In this case, the self-adjoint section B of End TM, characterized by g(BX, Y) = b(X, Y), will also be called a Codazzi tensor. The Codazzi tensor b will be called non-trivial if it is not a constant multiple of the metric. The aim of the present paper is to study some geometric and topological consequences of the existence of a non-trivial Codazzi tensor on a given Riemannian manifold. Results of this type were obtained by Bourguignon [3], who proved that the existence of such a tensor imposes strong restrictions on the curvature operator [3, Theoreme 5.1 and Corollaire 5.3] and, as a consequence, obtained the following theorem [3, Corollaire 7.3]: a compact orientable Riemannian four-manifold admitting a non-trivial Codazzi tensor with constant trace must have signature zero. Our main results consist in generalizing these theorems, in particular in seeing what can be said when the assumption on the trace is dropped. In § 2 of this paper we observe that, in the C°° category, every manifold admits a Riemannian metric with a non-trivial Codazzi tensor (Example 7), so that topological consequences may be expected only if some sort of analytic behaviour is assumed. Section 3 is devoted to the particular consequences of the existence of a non-trivial Codazzi tensor B for the structure of the curvature operator (Theorem \):for any point x of the manifold M and arbitrary eigenspaces Vx, V^ ofBx, the span Vx A V^cz A TXM of all exterior products of elements of Vx and V^ is invariant under the curvature operator Rx acting on 2-forms. As a consequence, we obtain in §4 a relation between the eigenspaces of any Codazzi tensor and the Pontryagin forms (Propositions 3 and 4), which, together with an extra argument for the case of a Codazzi tensor having only two distinct eigenvalues (Lemma 1), implies that a compact orientable Riemannian four-manifold (M, g) admitting a non-trivial Codazzi tensor b must have signature zero unless the restriction ofb to some non-empty open subset ofM is a constant multiple ofg (Theorem 2). Another consequence of Proposition 4 is that for any n-dimensional Riemannian manifold with a Codazzi tensor having n distinct eigenvalues almost everywhere, all the real Pontryagin classes are zero (Corollary 3).

79 citations

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TL;DR: In this paper, it was shown that if the Ricci curvature is uniformly bounded under the flow for all times $t\in [0,T)$, then the curvature tensor has to be uniformly bounded as well.
Abstract: Consider the unnormalized Ricci flow $(g_{ij})_t = -2R_{ij}$ for $t\in [0,T)$, where $T < \infty$. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times $t\in [0,T)$ then the solution can be extended beyond $T$. We prove that if the Ricci curvature is uniformly bounded under the flow for all times $t\in [0,T)$, then the curvature tensor has to be uniformly bounded as well.

78 citations

Journal ArticleDOI
TL;DR: In this paper, the wave operators do not factorize in general, and identify the Weyl tensor and its derivatives as the obstruction to factorization, and give a manifestly factorized form for them on (A)dS backgrounds for arbitrary spin and on Einstein backgrounds for spin 2.
Abstract: We analyze free conformal higher spin actions and the corresponding wave operators in arbitrary even dimensions and backgrounds. We show that the wave operators do not factorize in general, and identify the Weyl tensor and its derivatives as the obstruction to factorization. We give a manifestly factorized form for them on (A)dS backgrounds for arbitrary spin and on Einstein backgrounds for spin 2. We are also able to fix the conformal wave operator in d = 4 for s = 3 up to linear order in the Riemann tensor on generic Bach-flat backgrounds.

77 citations

Journal ArticleDOI
TL;DR: Owen et al. as discussed by the authors introduced the concept of tendex and vortex lines for visualizing spacetime curvature and applied it to weak-gravity phenomena, such as a spinning, gravitating point particle, two such particles side-by-side, a plane gravitational wave, and a point particle with a dynamical current-quadrupole moment.
Abstract: When one splits spacetime into space plus time, the Weyl curvature tensor (vacuum Riemann tensor) gets split into two spatial, symmetric, and trace-free tensors: (i) the Weyl tensor’s so-called electric part or tidal field Ɛ_(jk), which raises tides on the Earth’s oceans and drives geodesic deviation (the relative acceleration of two freely falling test particles separated by a spatial vector ξ^k is Δa_j=-Ɛ_(jk)ξ^k), and (ii) the Weyl tensor’s so-called magnetic part or (as we call it) frame-drag field B_(jk), which drives differential frame dragging (the precessional angular velocity of a gyroscope at the tip of ξ^k, as measured using a local inertial frame at the tail of ξ^k, is ΔΩ_j=B_(jk)ξ^k). Being symmetric and trace-free, Ɛ_(jk) and B_(jk) each have three orthogonal eigenvector fields which can be depicted by their integral curves. We call the integral curves of Ɛ_(jk)’s eigenvectors tidal tendex lines or simply tendex lines, we call each tendex line’s eigenvalue its tendicity, and we give the name tendex to a collection of tendex lines with large tendicity. The analogous quantities for B_(jk) are frame-drag vortex lines or simply vortex lines, their vorticities, and their vortexes. These concepts are powerful tools for visualizing spacetime curvature. We build up physical intuition into them by applying them to a variety of weak-gravity phenomena: a spinning, gravitating point particle, two such particles side-by-side, a plane gravitational wave, a point particle with a dynamical current-quadrupole moment or dynamical mass-quadrupole moment, and a slow-motion binary system made of nonspinning point particles. We show that a rotating current quadrupole has four rotating vortexes that sweep outward and backward like water streams from a rotating sprinkler. As they sweep, the vortexes acquire accompanying tendexes and thereby become outgoing current-quadrupole gravitational waves. We show similarly that a rotating mass quadrupole has four rotating, outward-and-backward sweeping tendexes that acquire accompanying vortexes as they sweep, and become outgoing mass-quadrupole gravitational waves. We show, further, that an oscillating current quadrupole ejects sequences of vortex loops that acquire accompanying tendex loops as they travel, and become current-quadrupole gravitational waves; and similarly for an oscillating mass quadrupole. And we show how a binary’s tendex lines transition, as one moves radially, from those of two static point particles in the deep near zone, to those of a single spherical body in the outer part of the near zone and inner part of the wave zone (where the binary’s mass monopole moment dominates), to those of a rotating quadrupole in the far wave zone (where the quadrupolar gravitational waves dominate). In Paper II we will use these vortex and tendex concepts to gain insight into the quasinormal modes of black holes, and in subsequent papers, by combining these concepts with numerical simulations, we will explore the nonlinear dynamics of curved spacetime around colliding black holes. We have published a brief overview of these applications in R. Owen et al. Phys. Rev. Lett. 106 151101 (2011). We expect these vortex and tendex concepts to become powerful tools for general relativity research in a variety of topics.

77 citations


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