Tangent Lie groups are Riemannian naturally reductive spaces

TL;DR

This paper proves that the tangent Lie group of a compact Lie group admits a naturally reductive Riemannian metric and a compatible metric connection with skew torsion, extending to certain almost tangent Lie groups.

Contribution

It establishes the existence of naturally reductive structures on tangent Lie groups and generalizes the construction to specific almost tangent Lie groups like $TS^7$.

Findings

Existence of left-invariant naturally reductive metrics on tangent Lie groups.

Construction of metric connections with skew torsion on these groups.

Extension of the structure to almost tangent Lie groups such as $TS^7$.

Abstract

Given a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, we consider its tangent Lie group $TG\cong G\ltimes_{\mathrm{Ad}} \mathfrak{g}$. In this short note, we prove that $TG$ admits a left-invariant naturally reductive Riemannian metric $g$ and a metric connection with skew torsion $\nabla$ such that $(TG,g,\nabla)$ is naturally reductive. An alternative spinorial description of the same connection on the direct product $G\times \mathfrak{g}$ generalizes in a rather subtle way to $TS^7$, which is in many senses almost a tangent Lie group.