Curvature of matrix and reductive Lie groups

TL;DR

This paper derives a simple formula for sectional curvatures on the general linear group and other matrix groups, exploring the geometric properties and their relation to commuting matrices.

Contribution

It provides a unified, simple formula for sectional curvatures on matrix groups and reductive Lie groups, linking curvature to matrix commutativity.

Findings

Derived a formula for sectional curvature on the general linear group.

Extended the curvature formula to reductive Lie groups.

Explored the connection between commuting matrices and zero curvature.

Abstract

In this paper, we give a simple formula for sectional curvatures on the general linear group, which is also valid for many other matrix groups. Similar formula is given for a reductive Lie group. We also discuss the relation between commuting matrices and zero sectional curvature.