Journal of Lie Theory 24 (2014), No. 3, 725--736
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.
roy-liu@utc.edu
Tin-Yau Tam
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, U.S.A.
tamtiny@auburn.edu
The concept of the t-geometric mean of two positive definite matrices is extended to symmetric spaces of noncompact type. The t-geometric 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 pre-order 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
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