Negatively curved Einstein metrics on Gromov-Thurston manifolds

TL;DR

This paper constructs infinitely many distinct closed manifolds in dimensions four and higher that support negatively curved Einstein metrics without being locally symmetric, expanding the known examples of such geometric structures.

Contribution

It provides the first infinite family of non-homotopic closed manifolds with negatively curved Einstein metrics that are not locally symmetric.

Findings

Existence of infinitely many such manifolds in each dimension ≥ 4.

These manifolds admit negatively curved Einstein metrics.

They do not admit locally symmetric metrics.

Abstract

For every $n\geq 4$ we construct infinitely many mutually not homotopic closed manifolds of dimension $n$ which admit a negatively curved Einstein metric but no locally symmetric metric.