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 Kähler-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.

Keywords

• levi-civita connection
• optical geometry
• congruence of shearfree geodesics
• sasaki manifolds