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

Abstract

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