The metrizability problem for Lorentz-invariant affine connections

Abstract

The invariant metrizability problem for affine connections on a manifold, formulated by Tanaka and Krupka for connected Lie groups actions, is considered in the particular cases of Lorentz and Poincaré (inhomogeneous Lorentz) groups. Conditions under which an affine connection on the open submanifold [Formula: see text] of the Euclidean space [Formula: see text] coincides with the Levi-Civita connection of some [Formula: see text], respectively [Formula: see text]-invariant metric field are studied. We give complete description of metrizable Lorentz-invariant connections. Explicit solutions (metric fields) of the invariant metrizability equations are found and their properties are discussed.