The Levi-Civita connections of Lorentzian manifolds with prescribed optical geometries

Abstract

We explicitly derive the Christoffel symbols in terms of adapted frame fields for the Levi-Civita connection of a Lorentzian n-manifold ( M, g ) , equipped with a prescribed optical geometry of Kahler-Sasaki type. The formulas found in this paper have several important applications, such as determining the geometric invariants of Lorentzian manifolds with prescribed optical geometries or solving curvature constraints.