On conformally recurrent manifolds of dimension greater than 4

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.