Recurrent Lorentzian Weyl spaces

Abstract

We find the local form of all non-closed Lorentzian Weyl manifolds $(M,c,\nabla)$ with recurrent curvature tensor.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. Two local structures are equivalent if and only if the corresponding functions are related by a transformation from $\mathrm{SAff}_1(\mathbb{R}) \times \mathrm{PSL}_2(\mathbb{R}) \times \mathbb{Z}_2$. 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 variables, 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 $\mathrm{Aff}(\mathbb R^3)$. 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.