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

TL;DR

This paper investigates 7-dimensional 3-Sasakian manifolds, identifying a special cocalibrated G_2 structure and its characteristic connection, which facilitate understanding their geometric properties despite the absence of a characteristic connection.

Contribution

It introduces a distinguished cocalibrated G_2 structure on 3-Sasakian manifolds and demonstrates how its characteristic connection and parallel spinor field can be used to analyze their geometry.

Findings

Identification of a special cocalibrated G_2 structure

Use of the characteristic connection and spinor for geometric analysis

Derivation of known and new properties of 3-Sasakian manifolds

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.