Negatively curved left-invariant metrics on Lie groups

摘要

We discuss negatively curved homogeneous spaces admitting a simply transitivegroup of isometries, or equivalently, negatively curved left-invariant metricson Lie groups. Negatively curved spaces have a remarkably rich and diversestructure and are interesting from both a mathematical and a physicalperspective. As well as giving general criteria for having left-invariantmetrics with negative Ricci curvature scalar, we also consider special cases,like Einstein spaces and Ricci nilsolitons. We point out the relevance thesespaces play in some higher-dimensional theories of gravity. In particular, weshow that the Ricci nilsolitons are Riemannian solutions to certainhigher-curvature gravity theories.