THE CHENG-YAU METRICS ON REGULAR CONVEX CONES AS HARMONIC IMMERSIONS INTO THE SYMMETRIC SPACE OF POSITIVE DEFINITE REAL SYMMETRIC MATRICES

概要

A Riemannian metric g on a domain Ω in R^n defines a map Fg from (Ω, g) into the symmetric space of positive definite real symmetric n×n matrices (Sym+(n), h), where h is the Cheng-Yau metric on Sym+(n). We show that the map Fg is a harmonic immersion if Ω is a regular convex cone and g is the Cheng-Yau metric on Ω. We also prove that the map Fg is totally geodesic if Ω is a homogeneous self-dual regular convex cone and g is the Cheng-Yau metric on Ω.