3-Sasakian manifolds in dimension seven, their spinors and G_2 structures

Abstract

It is well-known that 7-dimensional 3-Sasakian manifolds carry a one-parametric family of compatible G_2 structures and that they do not admit a characteristic connection. In this note, we show that there is nevertheless a distinguished cocalibrated G_2 structure in this family whose characteristic connection along with its parallel spinor field can be used for a thorough investigation of the geometric properties of 7-dimensional 3-Sasakian manifolds. Many known and some new properties can be easily derived from the properties of this connection and the spinor field, yielding thus an appropriate substitute for the missing characteristic connection.

9 pages

References in corpus (3)

• The Srni lectures on non-integrable geometries with torsion
• A brief note on the spectrum of the basic Dirac operator
• $G_2$-Manifolds With Parallel Characteristic Torsion

Cited by in corpus (22)

• Instantons and Killing spinors
• Spinorial description of $\mathrm{SU}(3)$- and $G_2$-manifolds
• Generalized Killing spinors and Lagrangian graphs
• Generalizations of 3-Sasakian manifolds and skew torsion
• Generalized Killing spinors on Einstein manifolds
• Einstein locally conformal calibrated $G_2$-structures
• Generalized Killing spinors on spheres
• Invariant Spinors on Homogeneous Spheres
• Killing and twistor spinors with torsion
• Quaternionic Heisenberg groups as naturally reductive homogeneous spaces
• Invariant connections and $\nabla$-Einstein structures on isotropy irreducible spaces
• Geometric Flows of $G_2$-Structures on 3-Sasakian 7-Manifolds
• Spinors of real type as polyforms and the generalized Killing equation
• Quaternionic Kaehler and Spin(7) metrics arising from quaternionic contact Einstein structures
• Jacobi relations on naturally reductive homogeneous spaces
• Homogeneous Sasakian and 3-Sasakian Structures from the Spinorial Viewpoint
• The second Dirac eigenvalue of a nearly parallel $G_2$-manifold
• Invariant affine connections on odd-dimensional spheres
• Revisiting 3-Sasakian and $G_2$-structures
• Cones of G manifolds and Killing spinors with skew torsion
• Einstein manifolds with skew torsion
• Families of solutions of the heterotic G${}_2$ system