There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with <i>n</i> Complete Light-Like Geodesics

Abstract

In the present paper, we study conformal mappings between a connected n-dimension pseudo-Riemannian Einstein manifolds. Let g be a pseudo-Riemannian Einstein metric of indefinite signature on a connected n-dimensional manifold M. Further assume that there is a point at which not all sectional curvatures are equal and through which in linearly independent directions pass n complete null (light-like) geodesics. If, for the function ψ the metric ψ − 2 g is also Einstein, then ψ is a constant, and conformal mapping is homothetic. Note that Kiosak and Matveev previously assumed that all light-lines were complete. If the Einstein manifold is closed, the completeness assumption can be omitted (the latter result is due to Mikeš and Kühnel).

Keywords

• pseudo-Riemannian manifold
• Einstein manifold
• concircular vector field
• conformal mapping
• light-like geodesic
• complete geodesic