On Metrizability of Invariant Affine Connections

TL;DR

This paper investigates when symmetric affine connections invariant under the rotation group SO(3) can be derived from invariant metrics, providing explicit solutions for the metrizability equations in Euclidean spaces.

Contribution

It offers a complete explicit characterization of SO(3)-invariant metrizable affine connections on Euclidean spaces, advancing understanding of symmetry-invariant geometric structures.

Findings

Explicit solutions for SO(3)-metrizability equations in 3D and 4D.

Characterization of G-metrizability conditions for invariant affine connections.

Complete description of all SO(3)-invariant metrizable connections.

Abstract

The metrizability problem for a symmetric affine connection on a manifold, invariant with respect to a group of diffeomorphisms G, is considered. We say that the connection is G-metrizable, if it is expressible as the Levi-Civita connection of a G-invariant metric field. In this paper we analyze the G-metrizability equations for the rotation group G = SO(3), acting canonically on three- and four-dimensional Euclidean spaces. We show that the property of the connection to be SO(3)-invariant allows us to find complete explicit description of all solutions of the SO(3)-metrizability equations.