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.

Six-dimensional supergravity and projective superfields

Abstract: We propose a superspace formulation of $ \mathcal{N} = \left( {1,0} \right) $ conformal supergravity in six dimensions. The corresponding superspace constraints are invariant under super-Weyl transformations generated by a real scalar parameter. The known variant Weyl super-multiplet is recovered by coupling the geometry to a super-3-form tensor multiplet. Isotwistor variables are introduced and used to define projective superfields. We formulate a locally supersymmetric and super-Weyl invariant action principle in projective superspace. Some families of dynamical supergravity-matter systems are presented.

Scattering amplitudes in super-renormalizable gravity

Abstract: We explicitly compute the tree-level on-shell four-graviton amplitudes in four, five and six dimensions for local and weakly nonlocal gravitational theories that are quadratic in both, the Ricci and scalar curvature with form factors of the d’Alembertian operator inserted between. More specifically we are interested in renormalizable, super-renormalizable or finite theories. The scattering amplitudes for these theories turn out to be the same as the ones of Einstein gravity regardless of the explicit form of the form factors. As a special case the four-graviton scattering amplitudes in Weyl conformal gravity are identically zero. Using a field redefinition, we prove that the outcome is correct for any number of external gravitons (on-shell n−point functions) and in any dimension for a large class of theories. However, when an operator quadratic in the Riemann tensor is added in any dimension (with the exception of the Gauss-Bonnet term in four dimensions) the result is completely altered, and the scattering amplitudes depend on all the form factors introduced in the action.

Curvature Estimates and the Positive Mass Theorem

Abstract: The Positive Mass Theorem implies that any smooth, complete, asymptotically flat3-manifold with non-negative scalar curvature which has zero total mass is isometricto (IR 3 ,δ ij ). In this paper, we quantify this statement using spinors and prove thatif a complete, asymptotically flat manifold with non-negative scalar curvature hassmall mass and bounded isoperimetric constant, then the manifold must be close to(IR 3 ,δ ij ), in the sense that there is an upper bound for the L 2 norm of the Riemanniancurvature tensor over the manifold except for a set of small measure. This curvatureestimate allows us to extend the case of equality of the Positive Mass Theorem toinclude non-smooth manifolds with generalized non-negative scalar curvature, whichwe define. 1 Introduction We introduce our problem in the context of General Relativity. Consider a 3 + 1 dimen-sional Lorentzian manifold N with metric g αβ of signature (− + ++). We denote theinduced Levi-Civita connection by ∇¯. Then the corresponding Ricci tensor R¯

Curvature collineations and the determination of the metric from the curvature in general relativity

Abstract: It is shown that for a very general class of space-times, the componentsR of the curvature tensor determine the metric components up to a constant conformal factor This general class contains most of those cases which are usually considered to be interesting from the point of view of Einstein's general relativity theory The connection between the above result and the existence of proper curvature collineations is given