The degree of mobility of Einstein metrics

TL;DR

This paper classifies the possible degrees of mobility for Einstein metrics in Riemannian and Lorentzian geometries, providing a comprehensive list of their projective equivalence classes and associated vector fields.

Contribution

It offers a complete classification of the degree of mobility for Einstein metrics on simply connected manifolds, detailing the structure of their projective symmetries.

Findings

Complete list of possible degrees of mobility for Einstein metrics.

Characterization of the space of essential projective vector fields.

Descriptions of projective equivalence classes for Einstein metrics.

Abstract

Two pseudo-Riemannian metrics are called projectively equivalent if their unparametrized geodesics coincide. The degree of mobility of a metric is the dimension of the space of metrics that are projectively equivalent to it. We give a complete list of possible values for the degree of mobility of Riemannian and Lorentzian Einstein metrics on simply connected manifolds, and describe all possible dimensions of the space of essential projective vector fields.