Recurrent Lorentzian Weyl Spaces

Abstract

We find the local form of all non-closed Lorentzian Weyl manifolds \((M,c,\nabla )\) with recurrent curvature tensor. The recurrent curvature tensor turns out to be weighted parallel, i.e., the obtained spaces provide certain generalization of locally symmetric affine spaces for the Weyl geometry. If the dimension of the manifold is greater than 3, then the conformal structure is flat, and the recurrent Weyl structure is locally determined by a single function of one variable. Two local structures are equivalent if and only if the corresponding functions are related by a transformation from \(\textrm{Aff}^0_1(\mathbb {R})\times \textrm{PSL}_2(\mathbb {R})\times {\mathbb {Z}}_2\). We find generators for the field of rational scalar differential invariants of this Lie group action. The global structure of the manifold M may be described in terms of a foliation with a transversal projective structure. It is shown that all locally homogeneous structures are locally equivalent, and there is only one simply connected homogeneous non-closed recurrent Lorentzian Weyl manifold. Moreover, there are 5 classes of cohomogeneity-one spaces, and all other spaces are of cohomogeneity-two. If \(\dim M=3\), the non-closed recurrent Lorentzian Weyl structures are locally determined by one function of two variables or two functions of one variable, depending on whether its holonomy algebra is 1- or 2-dimensional. In this case, two structures with the same holonomy algebra are locally equivalent if and only if they are related, respectively, by a transformation from an infinite-dimensional Lie pseudogroup or a 4-dimensional subgroup of \(\textrm{Aff}({\mathbb {R}}^3)\). Again we provide generators for the field of rational differential invariants. We find a local expression for the locally homogeneous non-closed recurrent Lorentzian Weyl manifolds of dimension 3, and also of those of cohomogeneity one and two. In the end we give a local description of the non-closed recurrent Lorentzian Weyl manifolds that are also Einstein–Weyl. All of them are 3-dimensional and have a 2-dimensional holonomy algebra.