Riemannian Geometry of Two Families of Tangent Lie Groups

TL;DR

This paper explores the Riemannian geometry of tangent bundles of specific Lie groups, analyzing connections and curvatures to understand their geometric properties.

Contribution

It provides a detailed study of Levi-Civita connection, sectional, and Ricci curvatures for tangent bundles of two particular Lie group families.

Findings

Explicit formulas for Levi-Civita connection on tangent Lie groups

Curvature properties of tangent bundles for special Lie groups

Insights into geometric structures of Lie groups with one-dimensional commutator

Abstract

Using vertical and complete lifts, any left invariant Riemannian metric on a Lie group induces a left invariant Riemannian metric on the tangent Lie group. In the present article we study the Riemannian geometry of tangent bundle of two families of Lie groups. The first one is the family of special Lie groups considered by J. Milnor and the second one is the class of Lie groups with one-dimensional commutator groups. The Levi-Civita connection, sectional and Ricci curvatures have been investigated.