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.

All Off-Shell R^2 Invariants in Five Dimensional N=2 Supergravity

Abstract: We construct supersymmetric completions of various curvature squared terms in five dimensional supergravity with eight supercharges. Adopting the dilaton Weyl multiplet, we obtain the minimal off-shell supersymmetric Ricci scalar squared as well as all vector multiplets coupled curvature squared invariants. Since the minimal off-shell supersymmetric Riemann tensor squared and Gauss-Bonnet combination in the dilaton Weyl multiplet have been obtained before, both the minimal off-shell and the vector multiplets coupled curvature squared invariants in the dilation Weyl multiplet are complete. We also constructed an off-shell Ricci scalar squared invariant utilizing the standard Weyl multiplet. The supersymmetric Ricci scalar squared in the standard Weyl multiplet is coupled to n number of vector multiplets by construction, and it deforms the very special geometry. We found that in the supersymmetric AdS_5 vacuum, the very special geometry defined on the moduli space is modified in a simple way. Finally, we study the magnetic string and electric black hole solutions in the presence of supersymmetric Ricci scalar squared.

An exotic t1s 4 with positive curvature

Abstract: We construct a metric with positive sectional curvature on a 7-manifold which supports an isometry group with orbits of codimension 1. It is a connection metric on the total space of an orbifold 3-sphere bundle over an orbifold 4-sphere. By a result of S. Goette, the manifold is homeomorphic but not diffeomorphic to the unit tangent bundle of the 4-sphere. Spaces of positive curvature play a special role in geometry. Although the class of manifolds with positive (sectional) curvature is expected to be relatively small, so far there are only a few known obstructions. Moreover, for closed simply connected manifolds these coincide with the known obstructions to non-negative curvature which are: (1) the Betti number theorem of Gromov which asserts that the homology of a compact manifold with non-negative sectional curvature has an a priori bound on the number of generators depending only on the dimension; and (2) a result of Lichnerowicz and Hitchin implying that a spin manifold with non-trivial ˆ A genus or generalized a genus cannot admit a metric with non-negative curvature. One way to gain further insight is to construct and analyze examples. This is quite difficult and has been achieved only a few times. Aside from the classical rank- one symmetric spaces, i.e. the spheres and the projective spaces with their canoni- cal metrics, and the recently proposed deformation of the so-called Gromoll-Meyer sphere (PW2), examples were only found in the '60s by Berger (Ber), in the '70s by Wallach (W) and by Aloff and Wallach (AlW), in the '80s by Eschenburg (E1,2), and in the '90s by Bazaikin (Ba). The examples by Berger, Wallach and Aloff-Wallach were shown, by Wallach in even dimensions (W) and by Berard Bergery (Be) in odd dimensions, to constitute a classification of simply connected homogeneous mani- folds of positive curvature, whereas the examples due to Eschenburg and Bazaikin typically are non-homogeneous, even up to homotopy. All of these examples can be obtained as quotients of compact Lie groups G with a bi-invariant metric by a free isometric "two sided" action of a subgroup H ⊂ G × G. Since a Lie group with a bi- invariant metric has non-negative curvature so do such quotients, and in rare cases

Massive graviton on arbitrary background: derivation, syzygies, applications

Abstract: We give the detailed derivation of the fully covariant form of the quadratic action and the derived linear equations of motion for a massive graviton in an arbitrary background metric (which were presented in arXiv:1410.8302 [hep-th]). Our starting point is the de Rham-Gabadadze-Tolley (dRGT) family of ghost free massive gravities and using a simple model of this family, we are able to express this action and these equations of motion in terms of a single metric in which the graviton propagates, hence removing in particular the need for a “reference metric' which is present in the non perturbative formulation. We show further how 5 covariant constraints can be obtained including one which leads to the tracelessness of the graviton on flat space-time and removes the Boulware-Deser ghost. This last constraint involves powers and combinations of the curvature of the background metric. The 5 constraints are obtained for a background metric which is unconstrained, i.e. which does not have to obey the background field equations. We then apply these results to the case of Einstein space-times, where we show that the 5 constraints become trivial, and Friedmann-Lemaitre-Robertson-Walker space-times, for which we correct in particular some results that appeared elsewhere. To reachmore » our results, we derive several non trivial identities, syzygies, involving the graviton fields, its derivatives and the background metric curvature. These identities have their own interest. We also discover that there exist backgrounds for which the dRGT equations cannot be unambiguously linearized.« less

Singular Spaces of Non-Positive Curvature

Abstract: In [Gr5] Gromov explains the definition of upper curvature bounds for singular spaces, a concept which goes back to A.D. Aleksandrov, cf. [ABN]. Below is a discussion of this material. The main application is a criterion for the hyperbolicity of certain simply connected polyhedra.

Instability of meridional axial system in f( R) gravity

Abstract: We analyze the dynamical instability of a non-static reflection axial stellar structure by taking into account the generalized Euler equation in metric f(R) gravity. Such an equation is obtained by contracting the Bianchi identities of the usual anisotropic and effective stress-energy tensors, which after using a radial perturbation technique gives a modified collapse equation. In the realm of the \(R+\epsilon R^n\) gravity model, we investigate instability constraints at Newtonian and post-Newtonian approximations. We find that the instability of a meridional axial self-gravitating system depends upon the static profile of the structure coefficients, while f(R) extra curvature terms induce the stability of the evolving celestial body.