Ricci decomposition

About: Ricci decomposition is a research topic. Over the lifetime, 1972 publications have been published within this topic receiving 45295 citations.

Cheeger-gromoll type metrics on the (1,1)-tensor bundles

Abstract: Using a Riemannian metric on a differentiable manifold, a Cheeger-Gromoll type metric is introduced on the (1,1)-tensor bundle of the manifold. Then the Levi-Civita connection, Riemannian curvature tensor, Ricci tensor, scalar curvature and sectional curvature of this metric are calculated. Also, a para-Nordenian structure on the the (1,1)-tensor bundle with this metric is constructed and the geometric properties of this structure are studied.

On the Ricci curvature of nonunimodular solvable metric Lie algebras of small dimension

Abstract: The Ricci curvature of solvable metric Lie algebras is studied. In particular, we prove that the Ricci operator of any metric nonunimodular solvable Lie algebra of dimension not exceeding 6 has at least two negative eigenvalues, that generalizes the known results.

Some higher-dimensional vacuum solutions

Abstract: We study an even-dimensional manifold with a pseudo-Riemannian metric with arbitrary signature and arbitrary dimensions. We consider the Ricci flat equations and present a procedure to construct solutions to some higher- (even-) dimensional Ricci flat field equations from the four-dimensional Ricci flat metrics. When the four-dimensional Ricci flat geometry corresponds to a colliding gravitational vacuum spacetime our approach provides an exact solution to the vacuum Einstein field equations for colliding gravitational plane waves in an (arbitrary) even-dimensional spacetime. We give explicitly higher-dimensional Szekeres metrics and study their singularity behaviour.

On the Ricci Curvature of Compact Spacelike Hypersurfaces in de Sitter Space

Abstract: In this paper we establish a sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a pinching condition for the Ricci curvature. Our result will be a consequence of an integral formula involving the Ricci curvature and the scalar curvature of the hypersurface. We also derive some other consequences and applications of this formula.

Poincaré‐Irreducible Tensor Operators for Positive‐Mass One‐Particle States. II

Abstract: The main objective of this article is the relativistic generalization of the ordinary SO(3)‐irreducible spin tensor operators for particles with positive mass. Two classes of relativistic one‐particle tensor operators are constructed. The tensor operators of the first class transform according to those representations of the Poincare group that are induced by the one‐valued unitary irreducible representations of the pseudo‐unitary group SU(1, 1) which belong to the continuous principal and the discrete principal series. These tensors are operator‐valued functions of a spacelike 4‐momentum transfer. The tensor operators of the second class correspond to vanishing 4‐momentum transfer. They transform according to those representations of the Poincare group that are induced by the unitary irreducible representations of the pseudo‐orthogonal group SO(3, 1) or its universal covering group SL(2C) which belong to the principal series. Both classes of Poincare‐irreducible tensor operators are constructed in a spin helicity basis for timelike 4‐momentum by means of projection operators which are continuous linear superpositions of unitary operator realizations for the groups SU(1, 1) and SL(2C). The Clebsch‐Gordan coefficients associated with the reduction into the two classes of Poincare‐irreducible tensor operators of a dyadic product of spin‐helicity basis vectors are calculated.