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

Alberto Dolcetti; Donato Pertici

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

Differential Geometry 2014-12-16 v1

Authors: Alberto Dolcetti , Donato Pertici

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 $SL_n \times \mathbb{R}$ .

Keywords

trace metric general linear group semi-Riemannian structure geodesics curvature foliation product manifold

Cite

@article{arxiv.1412.4565,
  title  = {Some differential properties of $GL_n(\mathbb{R})$ with the trace metric},
  author = {Alberto Dolcetti and Donato Pertici},
  journal= {arXiv preprint arXiv:1412.4565},
  year   = {2014}
}

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