Curvatures of metric Jordan algebras

TL;DR

This paper explores the geometric properties of metric Jordan algebras by defining a unique connection, introducing curvature tensors, and establishing existence results for specific metrics, including Jordan-Einstein metrics.

Contribution

It introduces the Jordan-Levi-Civita connection and derives curvature formulas, proving existence and non-existence results for certain metrics on Jordan algebras.

Findings

Every metric Jordan algebra has a unique Jordan-Levi-Civita connection.

Formally real Jordan algebras admit non-positive Jordan curvature and Jordan-Einstein metrics.

Nilpotent Jordan algebras do not admit Jordan-Einstein metrics.

Abstract

In this paper, we investigate metric Jordan algebras, and follow the lines of the paper (J. Milnor: Curvatures of left invariant metrics on Lie groups. Adv. Math. (1976)). Firstly, we define the Jordan-Levi-Civita connection, then we show that every metric Jordan algebra admits a unique Jordan-Levi-Civita connection. Secondly, using the Jordan-Levi-Civita connection, we introduce three natural curvature tensors on metric Jordan algebras, and obtain the corresponding curvature formulas. Thirdly, based on these curvature formulas, we prove that every formally real Jordan algebra admits both a metric of non-positive Jordan curvature, and a Jordan-Einstein metric of negative Jordan scalar curvature. Besides, for nilpotent Jordan algebras, we prove that they admit no Jordan-Einstein metrics.