Half-flat structures inducing Einstein metrics on homogeneous spaces

TL;DR

This paper studies special geometric structures called half-flat $SU(3)$-structures on homogeneous spaces, identifying conditions under which they induce Einstein metrics and providing specific examples and non-existence results.

Contribution

It characterizes the intrinsic torsion of half-flat structures and demonstrates the existence and non-existence of Einstein metrics induced by these structures on certain homogeneous spaces.

Findings

Existence of a half-flat structure inducing Einstein metric on $S^3\times S^3$

Non-existence of coupled structures inducing Einstein metrics on the same space

No coupled structures induce Einstein metrics on Einstein solvmanifolds

Abstract

In this paper, we consider half-flat $SU(3)$-structures and the subclasses of coupled and double structures. In the general case we show that the intrinsic torsion form $w_1^-$ is constant in each of the two subclasses. We then consider the problem of finding half-flat structures inducing Einstein metrics on homogeneous spaces. We give an example of a left invariant half-flat (non coupled and non double) structure inducing an Einstein metric on $S^3\times S^3$ and we show there does not exist any left invariant coupled structure inducing an ${\rm Ad}(S^1)$-invariant Einstein metric on it. Finally, we show that there are no coupled structures inducing the Einstein metric on Einstein solvmanifolds and on homogeneous Einstein manifolds of nonpositive sectional curvature.