
Journal of Lie Theory 24 (2014), No. 3, 725736 Copyright Heldermann Verlag 2014 A Geometric Mean for Symmetric Spaces of Noncompact Type Ming Liao Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, U.S.A. liaomin@auburn.edu Xuhua Liu Department of Mathematics, The University of Tennessee, Chattanooga, TN 37403, U.S.A. royliu@utc.edu TinYau Tam Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, U.S.A. tamtiny@auburn.edu The concept of the tgeometric mean of two positive definite matrices is extended to symmetric spaces of noncompact type. The tgeometric mean of two points in such a symmetric space yields the unique geodesic joining the points and the geometric mean is the midpoint. A parametrization of the geodesic in terms of the two points is given. Inequalities about geometric mean and geodesic triangle are given in terms of Kostant's preorder on semisimple Lie groups as well as on their Lie algebras. Keywords: Geometric mean, positive definite matrices, symmetric spaces, semisimple Lie groups, geodesics, log majorization, Kostant's order. MSC: 15A45, 15A48, 53C35 [ Fulltextpdf (280 KB)] for subscribers only. 