**Alberto Dolcetti**^{[1]} and **Donato Pertici**^{[2]}

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

**Pages:** 267-286

**Received:** 24 October 2014

**Accepted in revised form:** 3 February 3 2015

**Mathematics Subject Classification (2010):** 53C50, 53C22, 15A16.

**Keywords:** Trace metric, Semi-Riemannian manifolds, geodesics, curvature tensors, nonsingular (special) matrices,
exponential and logarithm of a real matrices.

**Author address:**

[1], [2] : Dipartimento di Matematica e Informatica "Ulisse Dini",Universitá degli Studi di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy

**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 \(SL_n \times \mathbb{R}\).

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