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.

Ricci curvature bounds and Einstein metrics on compact manifolds

Abstract: where f: MO M1 is a homeomorphism and dil f is the dilatation of f given by dil f = supX,X2 dist(f(x) , f(x2))/ dist(x1, x2) . If MO and M1 are not homeomorphic, define dL(MO, MI) = +oo. Gromov [20] proves the remarkable result that the space of compact Riemannian manifolds f((A, 3, D) of sectional curvature IKI 3 > 0, and diameter dM v, and diameter dM c(IKI , dm, VM1) In particular, Gromov's compactness theorem may be strengthened to the statement that f((A, v , D) is C1 'l compact in the Lipschitz topology. In this paper, we study the question of Lipschitz convergence of compact Riemannian manifolds with bounds imposed on the Ricci curvature Ric in

Covariant conservation of energy momentum in modified gravities

Abstract: An explicit proof of the vanishing of the covariant divergence of the energy-momentum tensor in modified theories of gravity is presented. The gravitational action is written in arbitrary dimensions and allowed to depend nonlinearly on the curvature scalar and its couplings with a scalar field. Also the case of a function of the curvature scalar multiplying a matter Lagrangian is considered. The proof is given both in the metric and in the first-order formalism, i.e. under the Palatini variational principle. It is found that the covariant conservation of energy-momentum is built-in to the field equations. This crucial result, called the generalized Bianchi identity, can also be deduced directly from the covariance of the extended gravitational action. Furthermore, we demonstrate that in all of these cases, the freely falling world lines are determined by the field equations alone and turn out to be the geodesics associated with the metric compatible connection. The independent connection in the Palatini formulation of these generalized theories does not have a similar direct physical interpretation. However, in the conformal Einstein frame a certain bi-metricity emerges into the structure of these theories. In the light of our interpretation of the independent connection as an auxiliary variable we can also reconsider some criticisms of the Palatini formulation originally raised by Buchdahl.

Generalized entropy and higher derivative gravity

Abstract: We derive an extension of the Ryu-Takayanagi prescription for curvature squared theories of gravity in the bulk, and comment on a prescription for more general theories. This results in a new entangling functional, that contains a correction to Wald’s entropy. The new term is quadratic in the extrinsic curvature. The coefficient of this correction is a second derivative of the lagrangian with respect to the Riemann tensor. For Gauss-Bonnet gravity, the new functional reduces to Jacobson-Myers’.

Hamiltonian formulation of the teleparallel description of general relativity

Abstract: The Hamiltonian formulation of the teleparallel description of Einstein’s general relativity is established. Under a particular gauge fixing the Hamiltonian of the theory is written in terms of first class constraints. The algebra of the Hamiltonian and vector constraints resembles that of the standard Arnowitt–Deser–Misner formulation. This geometrical framework might be relevant as it is known that in manifolds with vanishing curvature tensor but with nonzero torsion tensor it is possible to carry out a simple construction of Becchi–Rouet–Stora–Tyutin‐like operators.

Curvature Collineations: A Fundamental Symmetry Property of the Space‐Times of General Relativity Defined by the Vanishing Lie Derivative of the Riemann Curvature Tensor

Abstract: A Riemannian space Vn is said to admit a particular symmetry which we call a ``curvature collineation'' (CC) if there exists a vector ξi for which £ξRjkmi=0, where Rjkmi is the Riemann curvature tensor and £ξ denotes the Lie derivative. The investigation of this symmetry property of space‐time is strongly motivated by the all‐important role of the Riemannian curvature tensor in the theory of general relativity. For space‐times with zero Ricci tensor, it follows that the more familiar symmetries such as projective and conformal collineations (including affine collineations, motions, conformal and homothetic motions) are subcases of CC. In a V4 with vanishing scalar curvature R, a covariant conservation law generator is obtained as a consequence of the existence of a CC. This generator is shown to be directly related to a generator obtained by means of a direct construction by Sachs for null electromagnetic radiation fields. For pure null‐gravitational space‐times (implying vanishing Ricci tensor) which admit CC, a similar covariant conservation law generator is shown to exist. In addition it is found that such space‐times admit the more general generator (recently mentioned by Komar for the case of Killing vectors) of the form (−g Tijkmξiξjξk);m=0, involving the Bel‐Robinson tensor Tijkm. Also it is found that the identity of Komar, [−g(ξi;j−ξj;i)];i;j=0, which serves as a covariant generator of field conservation laws in the theory of general relativity appears in a natural manner as an essentially trivial necessary condition for the existence of a CC in space‐time. In addition it is shown that for a particular class of CC,£ξK is proportional to K, where K is the Riemannian curvature defined at a point in terms of two vectors, one of which is the CC vector. It is also shown that a space‐time which admits certain types of CC also admits cubic first integrals for mass particles with geodesic trajectories. Finally, a class of null electromagnetic space‐times is analyzed in detail to obtain the explicit CC vectors which they admit.