Related papers: On conformally recurrent manifolds of dimension gr…

Conformally quasi-recurrent (CQR)_n pseudo-Riemannian manifolds are investigated, and several new results are obtained. It is shown that the Ricci tensor and the gradient of the fundamental vector are Weyl compatible tensors (the notion was…

It is shown that in every dimension n=3j+2, j=1,2,3,..., there exist compact pseudo-Riemannian manifolds with parallel Weyl tensor, which are Ricci-recurrent, but neither conformally flat nor locally symmetric, and represent all indefinite…

This paper derives new identities for the Weyl tensor on a gradient Ricci soliton, particularly in dimension four. First, we prove a Bochner-Weitzenb\"ock type formula for the norm of the self-dual Weyl tensor and discuss its applications,…

We discuss the algebraic classification of the Weyl tensor in higher dimensional Lorentzian manifolds. This is done by characterizing algebraically special Weyl tensors by means of the existence of aligned null vectors of various orders of…

We prove the existence of compact pseudo-Riemannian manifolds with parallel Weyl tensor which are neither conformally flat nor locally symmetric, and represent all indefinite metric signatures in all dimensions $\,n\ge5$. Until now such…

Let N be a symmetric space of dimension n > 5 whose de Rham decomposition contains no factors of constant curvature and let W be the Weyl tensor of N at some point. We prove that a Riemannian manifold whose Weyl tensor at every point is a…

In this article, we give a new proof of a result due to J. Kim, which states that the Ricci tensor of a gradient Ricci soliton with dimension $n \geq 4$ and harmonic Weyl tensor has at most three distinct eigenvalues. This result…

The rotations of rigid bodies in Euclidean space are characterized by their instantaneous angular velocity and angular momentum. In an arbitrary number of spatial dimensions, these quantities are represented by bivectors (antisymmetric…

A Riemannian manifold is called Weyl homogeneous, if its Weyl tensors at any two points are "the same", up to a positive multiple. A Weyl homogeneous manifold is modeled on a homogeneous space $M_0$, if its Weyl tensor at every point is…

Compact pseudo-Riemannian manifolds that have parallel Weyl tensor without being conformally flat or locally symmetric are known to exist in infinitely many dimensions greater than 4. We prove some general topological properties of such…

We introduce the new algebraic property of Weyl compatibility for symmetric tensors and vectors. It is strictly related to Riemann compatibility, which generalizes the Codazzi condition while preserving much of its geometric implications.…

On a (pseudo-) Riemannian manifold of dimension n > 2, the space of tensors which transform covariantly under Weyl rescalings of the metric is built. This construction is related to a Weyl-covariant operator D whose commutator [D,D] gives…

We determine the local structure of all pseudo-Riemannian manifolds $(M,g)$ in dimensions $n\ge4$ whose Weyl conformal tensor $W$ is parallel and has rank 1 when treated as an operator acting on exterior 2-forms at each point. If one fixes…

The peeling behaviour of the Weyl tensor near null infinity is determined for an asymptotically flat higher dimensional spacetime. The result is qualitatively different from the peeling property in 4d. To leading order, the Weyl tensor is…

We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the…

We explore connections between geometrical properties of null congruences and the algebraic structure of the Weyl tensor in n>4 spacetime dimensions. First, we present the full set of Ricci identities on a suitable "null" frame, thus…

We develop a dimension-independent theory of alignment in Lorentzian geometry, and apply it to the tensor classification problem for the Weyl and Ricci tensors. First, we show that the alignment condition is equivalent to the PND equation.…

We prove that, in a space-time of dimension n>3 with a velocity field that is shear-free, vorticity-free and acceleration-free, the covariant divergence of the Weyl tensor is zero if the contraction of the Weyl tensor with the velocity is…

We prove theorems about the Ricci and the Weyl tensors on generalized Robertson-Walker space-times of dimension $n\ge 3$. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid…

Irreducible bilinear tensorial concomitants of an arbitrary complex antisymmetric valence-2 tensor are derived in four-dimensional spacetime. In addition these bilinear concomitants are symmetric (or antisymmetric), self-dual (or…