Geodesics for Dual Connections and Means on Symmetric Cones

Abstract.

Means on self-dual and homogeneous cones (, i.e., symmetric cones) are discussed from a viewpoint of differential geometry with affine connections. We first define means on symmetric cones in an axiomatic way following [8]. Next we consider dualistic differential geometry (, i.e., Riemannian metric and affine connections) [1] naturally introduced on symmetric cones. Elucidating the relation between the geodesics defined by each affine connection, and operator monotone functions that generate means, we show an important class of means are expressed by the (mid)points on geodesics. Related results on the means and submanifolds in a symmetric cone are also presented.