Skewable matrices over $\wedge^2 V$ applied to locally metric connections

TL;DR

This paper presents an algorithm to determine when a curvature matrix of 2-forms in a vector bundle is skew symmetric, aiding in identifying locally metric connections, with applications to curvature analysis.

Contribution

It introduces a simple algorithm for testing skew symmetry of 2-form matrices, facilitating the verification of locally metric connections in vector bundles.

Findings

Algorithm effectively identifies skew symmetric curvature matrices.

Application to full rank curvature connections confirms local metricity.

Provides a practical tool for geometric analysis of connections.

Abstract

A necessary condition for a connection in a vector bundle to be locally metric is for its curvature matrix, which consists of $2$ forms, to be skew symmetric with respect to some local frame. In this paper we give a simple algorithm that can be used to decide when a matrix of $2$ forms is equivalent to a skew symmetric matrix. We apply this algorithm to verify whether a "{\it full rank}" curvature connection is locally metric.