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

摘要

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