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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 paper, the Euler-Poincar equation associated with the geodesic flow of the right invariant metric on a compact Riemannian manifold was generalized to the setting of an n-dimensional compact manifold.
Abstract: Recently, Holm, Marsden, and Ratiu [1998] have derived a new model for the mean motion of an ideal fluid in Euclidean space given by the equation $\dot{V}(t) + abla_{U(t)} V(t) - \alpha^2 [ abla U(t)]^t \cdot \triangle U(t) = -\text{grad} p(t)$ where $\text{div} U=0$, and $V = (1- \alpha^2 \triangle)U$. In this model, the momentum $V$ is transported by the velocity $U$, with the effect that nonlinear interaction between modes corresponding to length scales smaller than $\alpha$ is negligible. We generalize this equation to the setting of an $n$ dimensional compact Riemannian manifold. The resulting equation is the Euler-Poincar\'{e} equation associated with the geodesic flow of the $H^1$ right invariant metric on ${\mathcal D}^s_\mu$, the group of volume preserving Hilbert diffeomorphisms of class $H^s$. We prove that the geodesic spray is continuously differentiable from $T{\mathcal D}_\mu^s(M)$ into $TT{\mathcal D}_\mu^s(M)$ so that a standard Picard iteration argument proves existence and uniqueness on a finite time interval. Our goal in this paper is to establish the foundations for Lagrangian stability analysis following Arnold [1966]. To do so, we use submanifold geometry, and prove that the weak curvature tensor of the right invariant $H^1$ metric on ${\mathcal D}^s_\mu$ is a bounded trilinear map in the $H^s$ topology, from which it follows that solutions to Jacobi's equation exist. Using such solutions, we are able to study the infinitesimal stability behavior of geodesics.

128 citations

Journal ArticleDOI
TL;DR: For general fourth-order theories, described by actions which are general functions of the scalar curvature, the Ricci tensor and the full Riemann tensor, it is shown that the higher-derivative theories may have multiple stable vacua.
Abstract: A discussion of an extended class of higher-derivative classical theories of gravity is presented. A procedure is given for exhibiting the new propagating degrees of freedom, at the full nonlinear level, by transforming the higher-derivative action to a canonical second-order form. For general fourth-order theories, described by actions which are general functions of the scalar curvature, the Ricci tensor and the full Riemann tensor, it is shown that the higher-derivative theories may have multiple stable vacua. The vacua are shown to be, in general, nontrivial, corresponding to de Sitter or anti-de Sitter solutions of the original theory. It is also shown that around any vacuum the elementary excitations remain the massless graviton, a massive scalar field, and a massive ghostlike spin-two field. The analysis is extended to actions which are arbitrary functions of terms of the form ${\ensuremath{ abla}}^{2k}R$, and it is shown that such theories also have a nontrivial vacuum structure.

127 citations

Journal ArticleDOI
TL;DR: In this article, a general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed, which is characterized by the tensors of Riemanian and non-Riemannians curvatures, an affine deformation tensor being the result of nonmetricity of one of the connections.
Abstract: A general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed. It is shown that the averaged manifold has a metric and two equi-affine symmetric connections. The geometry of the space is characterized by the tensors of Riemannian and non-Riemannian curvatures, an affine deformation tensor being the result of non-metricity of one of the connections. To average out the differential Bianchi identities, correlation 2-form, 3-form and 4-form are introduced and the differential relations on these correlations tensors are derived, the relations being integrable on an arbitrary averaged manifold. Upon assuming a splitting rule for the average of the product including a covariantly constant tensor, an averaging out of the Einstein equations has been carried out which brings additional terms with the correlation tensors into them. As shown by averaging out the contracted Bianchi identities, the equations of motion for the averaged energy-momentum tensor do also include the geometric correction terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (then the non-Riemannian one is the macroscopic gravitational field), a theorem that relates the algebraic structure of the averaged microscopic metric with that of the induction tensor is proved. Due to the theorem the same field operator as in the Einstein equations is manifestly extracted from the averaged ones. Physical interpretation and application of the relations and equations obtained to treat macroscopic gravity are discussed.

127 citations

Journal ArticleDOI
TL;DR: In this paper, a parity-violating contribution to the complete action which is linear in the curvature tensor and vanishes identically in the absence of torsion is shown.
Abstract: The general structure of metric-torsion theories of gravitation is shown to allow a parity-violating contribution to the complete action which is linear in the curvature tensor and vanishes identically in the absence of torsion. The resulting action involves apart from the Newtonian constant a coupling which governs the strength of the predicted parity-nonconserving ''interactions'' mediated by torsion. We consider this theory in the presence of the Proca field and show that it leads to a parity-violating term in the field equations in contrast to the Einstein-Cartan-Sciama-Kibble theory, which we use as a particularly simple example of a metric-torsion theory of gravitation.

126 citations


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