Metric connections with parallel skew-symmetric torsion

Abstract

A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel ${\mathrm{G}}_2$-manifolds, Sasakian and $3$-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds carrying a metric connection with parallel skew-symmetric torsion. On every such manifold one can define a natural splitting of the tangent bundle which gives rise to a Riemannian submersion over a geometry with parallel skew-symmetric torsion of smaller dimension endowed with some extra structure. We show how previously known examples of geometries with parallel skew-symmetric torsion fit into this pattern, and construct several new examples. In the particular case where the above Riemannian submersion has the structure of a principal bundle, we give the complete local classification of the corresponding geometries with parallel skew-symmetric torsion.

42 pages; thoroughly revised version, including a simpler definition of the geometry with parallel curvature determined by a geometry with parallel skew-symmetric torsion, and an appendix discussing 3-(α,δ)-Sasakian structures in our framework

References in corpus (5)

• New G2 holonomy cones and exotic nearly Kaehler structures on the 6-sphere and the product of a pair of 3-spheres
• Almost Hermitian Structures with Parallel Torsion
• Generalizations of 3-Sasakian manifolds and skew torsion
• $G_2$-Manifolds With Parallel Characteristic Torsion
• Sp(n)U(1)-connections with parallel totally skew-symmetric torsion

Cited by in corpus (7)

• Infinite families of homogeneous Bismut Ricci flat manifolds
• Homogeneous non-degenerate $3$-$(\alpha ,\delta )$-Sasaki manifolds and submersions over quaternionic Kähler spaces
• Affine Connections on 3-Sasakian Homogeneous Manifolds
• Canonical Submersions in Nearly Kähler Geometry
• ${\mathrm{G}}_2$-structures with parallel skew-symmetric torsion
• Bismut Ricci flat manifolds with symmetries
• Lorentzian connections with parallel twistor-free torsion