Related papers: The metric structure of compact rank-one ECS manif…
We study ECS manifolds, that is, pseudo-Riemannian manifolds with parallel Weyl tensor which are neither conformally flat nor locally symmetric. Every ECS manifold has rank 1 or 2, the rank being the dimension of a distinguished null…
Pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as ECS manifolds, have a natural local invariant, the rank, which equals 1 or 2, and is the dimension of a certain…
By ECS manifolds one means pseudo-Riemannian manifolds of dimensions $\,n\ge4\,$ which have parallel Weyl tensor, but not for one of the two obvious reasons: conformal flatness or local symmetry. As shown by Roter [10, 2], they exist for…
We construct new examples of compact ECS manifolds, that is, of pseudo-Riemannian manifolds with parallel Weyl tensor that are neither conformally flat nor locally symmetric. Every ECS manifold has rank 1 or 2, the rank being the dimension…
This is an exposition of recent results -- obtained in joint work with Andrzej Derdzinski -- on essentially conformally symmetric (ECS) manifolds, that is, those pseudo-Riemannian manifolds with parallel Weyl curvature which are not locally…
This is an exposition of some recent results on ECS manifolds, by which we mean pseudo-Riemannian manifolds of dimensions greater than 3 that are neither conformally flat nor locally symmetric, and have parallel Weyl tensor. All ECS metrics…
A three-dimensional pseudo-Riemannian manifold is called essentially conformally symmetric (ECS) if its Cotton tensor is parallel but nowhere-vanishing. In this note we prove that three-dimensional ECS manifolds must be noncompact or,…
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…
We compute the full holonomy group of compact Lorentzian manifolds with parallel Weyl tensor, which are neither conformally flat nor locally symmetric, for the case where the fundamental group is contained in a distinguished subgroup G of…
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…
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…
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…
A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. Osserman…
This is a final step in a local classification of pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric.
A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a sum of symmetric outer product of vectors. A rank-1 order-k…
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…
In this article, we study quasi-Einstein manifolds with constant scalar curvature. We provide a classification of compact and noncompact (possibly with boundary) $T$-flat quasi-Einstein manifolds with constant scalar curvature, where the…
The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z.I. Szabo for harmonic manifolds with compact universal cover. E. Damek and F. Ricci…
On any manifold, any non-degenerate symmetric 2-form (metric) and any skew-symmetric (differential) form W can be reduced to a canonical form at any point, but not in any neighborhood: the respective obstructions being the Riemannian tensor…
We introduce the concept of a Clifford-Weyl structure on a conformal manifold, which consists of an even Clifford structure parallel with respect to the tensor product of a metric connection on the Clifford bundle and a Weyl structure on…