Riv. Mat. Univ. Parma, Vol. 6, No. 2, 2015

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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