The Cheng-Yau metrics on regular convex cones as harmonic immersions into the symmetric space of positive definite real symmetric matrices

Abstract

A Riemannian metric $g$ on a domain $\Omega$ in $ \mathbb{R}^n$ defines a map $F_g$ from $(\Omega,g)$ into the symmetric space of positive definite real symmetric $n \times n$ matrices $(\textrm{Sym}^+(n),h)$, where $h$ is the Cheng-Yau metric on $\textrm{Sym}^+(n)$. 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 $\Omega$.