Rigidity of quasi-Einstein metrics: The incompressible case

TL;DR

This paper investigates divergence-free quasi-Einstein metrics on closed manifolds, revealing a rigidity property that these metrics always admit a one-parameter isometry group generated by the vector field.

Contribution

It characterizes the structure of divergence-free quasi-Einstein metrics, showing they inherently possess a one-parameter isometry group and establishing new geometric restrictions.

Findings

Metrics admit a one-parameter isometry group

Restrictions on topology and geometry of the spaces

Applicable to near-horizon geometries of black holes

Abstract

As part of a programme to classify quasi-Einstein metrics $(M,g,X)$ on closed manifolds and near-horizon geometries of extreme black holes, we study such spaces when the vector field $X$ is divergence-free but not identically zero. This condition is satisfied by left-invariant quasi-Einstein metrics on compact homogeneous spaces (including the near-horizon geometry of an extreme Myers-Perry black hole with equal angular momenta in two distinct planes), and on certain bundles over K\"ahler-Einstein manifolds. We find that these spaces exhibit a mild form of rigidity: they always admit a one-parameter group of isometries generated by $X$. Further geometrical and topological restrictions are also obtained.