Some differential properties of $GL_n(\mathbb{R})$ with the trace metric

TL;DR

This paper investigates the geometric properties of the general linear group over real numbers equipped with the trace metric, focusing on geodesics, curvature, and foliation structures.

Contribution

It provides new insights into the geometric structure of $GL_n(\mathbb{R})$, including its foliation by isometric leaves and the product structure of its positive determinant component.

Findings

Characterization of geodesics and curvature tensors

Existence of a foliation with leaves isometric to $SL_n(\mathbb{R})$

Component with positive determinant is isometric to $SL_n \times \mathbb{R}$

Abstract

In this note we consider some properties of $GL_n(\mathbb{R})$ with the Semi-Riemannian structure induced by the trace metric $g$. In particular we study geodesics and curvature tensors. Moreover we prove that $GL_n$ has a suitable foliation, whose leaves are isometric to $(SL_n(\mathbb{R}), g)$, while its component of matrices with positive determinant is isometric to the Semi-Riemannian product manifold $SL_n \times \mathbb{R}$.