Einstein manifolds with skew torsion

TL;DR

This paper systematically investigates Einstein manifolds with skew symmetric torsion, deriving their properties, conditions, and providing numerous examples including special geometric structures like Sasakian and G2 manifolds.

Contribution

It introduces the first systematic study of Einstein manifolds with skew torsion, deriving key equations, and constructing diverse examples across different geometric frameworks.

Findings

Einstein manifolds with parallel skew torsion have constant scalar curvature.

Complete positive scalar curvature cases are compact with finite fundamental group.

Examples include Einstein-Sasaki and 3-Sasakian manifolds with deformations into Einstein metrics with skew torsion.

Abstract

This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection with skew symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any Einstein manifold with skew torsion has constant scalar curvature; and if it is complete of positive scalar curvature, it is necessarily compact and it has finite first fundamental group. The longest part of the paper is devoted to the systematic construction of large families of examples. We discuss when a Riemannian Einstein manifold can be Einstein with skew torsion. We give examples of almost Hermitian, almost metric contact, and G2 manifolds that are Einstein with skew torsion. For example, we prove that any Einstein-Sasaki manifold and any 7-dimensional 3-Sasakian manifolds admit deformations into an Einstein metric with parallel skew torsion.