Classification of Left Invariant Riemannian metrics on Complex hyperbolic space

TL;DR

This paper classifies all left invariant Riemannian metrics on complex hyperbolic space, revealing their constant negative scalar curvature and identifying the unique Einstein metric among them, while exploring connections to Ricci solitons.

Contribution

It provides a complete classification of left invariant metrics on complex hyperbolic space and identifies the unique Einstein metric among them.

Findings

All metrics have constant negative scalar curvature.

Only one metric is Einstein (up to isometry and scaling).

Established relations between Ricci solitons on Heisenberg group and Einstein metrics.

Abstract

It is well known that $\mathbb{C}H^n$ has the structure of solvable Lie group with left invariant metric of constant holomorphic sectional curvature. In this paper we give the full classification of all possible left invariant Riemannian metrics on this Lie group. We prove that all of these metrics are of constant negative scalar curvature and only one of them is Einstein (up to isometry and scaling). Finally, we present the relation between Ricci solitons on Heisenberg group and Einstein metric on $\mathbb{C}H^n$.