Geodesics and curvature of semidirect product groups

TL;DR

This paper investigates the geometric properties, specifically geodesics and curvature, of semidirect product groups with right invariant metrics, providing explicit formulas and examples such as magnetic extensions.

Contribution

It derives formulas for geodesics and curvature of semidirect product groups, including special cases like isometric extensions, and explores examples like magnetic extensions.

Findings

Curvature of isometric semidirect products equals the sum of component curvatures.

Explicit formulas for geodesics in semidirect product groups.

Examples include magnetic extensions of groups.

Abstract

Geodesics and curvature of semidirect product groups with right invariant metrics are determined. In the special case of an isometric semidirect product, the curvature is shown to be the sum of the curvature of the two groups. A series of examples, like the magnetic extension of a group, are then considered.