Abstract

July 1 - 14.30-15.20
A.Ohara - Dualistic Differential Geometry on Symmetric Cones and its Applications

We discuss dually flat structures on symmetric (i.e., homogeneous and self-dual ) cones associated with Euclidean Jordan algebra. First we exploit relations between dual connections on symmetric cones and Euclidean Jordan algebras. In particular, we introduce the property called "doubly autoparallelism" and show how doubly autoparallel submanifolds are characterized by Jordan subalgebras. Next we define means on symmetric cones in an axiomatic way following Kubo-Ando theory and then we discuss them from a viewpoint of dualistic differential structure. We show that various means are expressed by the midpoints on geodesics with respect to the corresponding dualistic structures by elucidating the relation between the geodesics and operator monotone functions that generate means. [an error occurred while processing this directive]