Metric properties of domains in real-type Nagano spaces

Abstract

Nagano spaces are compact symmetric spaces that admit large transformation groups. They include for instance all the Grassmannians and the Einstein Universes. In this paper, we study a Kobayashi-type pseudometric on domains in real-type Nagano spaces. When the Nagano space is real projective space, this metric coincides with the classical Kobayashi pseudometric. For a dually convex domain of a general real-type Nagano space, we prove that this pseudometric is a genuine metric if and only if the domain does not contain a photon minus a point. We compute this metric on the proper symmetric domains and prove that it is obtained by integrating the $L^1$-norm along flats. We prove that in higher rank, the Kobayashi metric of a strongly $\mathcal{R}$-proper dually convex divisible domain is never Gromov hyperbolic. This contrasts with the rank-one case corresponding to real projective space, where a classical result of Benoist shows that this metric is Gromov hyperbolic if and only if the domain is strictly convex.