Some differential properties of GL_n(R) with the trace metric

In this note we consider some properties of GLn(R)GLn(R) with the Semi-Riemannian structure induced by the trace metric gg. In particular we study geodesics and curvature tensors. Moreover we prove that GLnGLn has a suitable foliation, whose leaves are isometric to (SLn(R),g)(SLn(R),g), while its component of matrices with positive determinant is isometric to the Semi-Riemannian product manifold SLn×RSLn×R.