A note on locally metric connections

TL;DR

This paper provides necessary and sufficient local conditions for a connection in a plane bundle over a surface to be metric, along with a global criterion for metric equivalence based on Euler class.

Contribution

It introduces easily verifiable local criteria for locally metric connections and a global condition relating Euler class to metric equivalence.

Findings

Derived necessary and sufficient local conditions for locally metric connections.

Established a global criterion for metric equivalence using Euler class.

Provided practical verification methods in local charts.

Abstract

In this paper we give necessary and sufficient conditions for a connection in a plane bundle above a surface to be locally metric. These conditions are easy to be verified in any local chart. Also as a global result we give a necessary condition for two connections to be metric equivalent in terms of their Euler class.