A Geometric Mean for Symmetric Spaces of Noncompact Type

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.