On conformally recurrent manifolds of dimension greater than 4

TL;DR

This paper studies conformally recurrent pseudo-Riemannian manifolds of dimension greater than 4, deriving explicit forms of the Ricci tensor and classifying Lorentzian cases based on the Weyl tensor's properties.

Contribution

It provides explicit expressions for the Ricci tensor and classifies Lorentzian conformally recurrent manifolds based on the Weyl tensor's algebraic type.

Findings

Ricci tensor has at most two distinct eigenvalues.

Manifolds with nonzero Weyl tensor square are decomposable.

Weyl tensor type depends on the recurrence vector's nature.

Abstract

Conformally recurrent pseudo-Riemannian manifolds of dimension n>4 are investigated. The Weyl tensor is represented as a Kulkarni-Nomizu product. If the square of the Weyl tensor is nonzero, a covariantly constant symmetric tensor is constructed, that is quadratic in the Weyl tensor. Then, by Grycak's theorem, the explicit expression of the traceless part of the Ricci tensor is obtained, up to a scalar function. The Ricci tensor has at most two distinct eigenvalues, and the recurrence vector is an eigenvector. Lorentzian conformally recurrent manifolds are then considered. If the square of the Weyl tensor is nonzero, the manifold is decomposable. A null recurrence vector makes the Weyl tensor of algebraic type IId or higher in the Bel - Debever - Ortaggio classification, while a time-like recurrence vector makes the Weyl tensor purely electric.