Homogeneous Sasakian and 3-Sasakian Structures from the Spinorial Viewpoint

TL;DR

This paper presents a spinorial approach to constructing homogeneous Sasakian and 3-Sasakian structures in arbitrary dimensions, extending known results and providing a detailed description of invariant spinors and Killing spinors on these spaces.

Contribution

It generalizes the construction of Sasakian and 3-Sasakian structures via spinors to arbitrary dimensions and characterizes invariant spinors and Killing spinors on homogeneous spaces.

Findings

Complete description of invariant spinors on 3-Sasakian spaces

Basis for invariant Riemannian Killing spinors

Identification of spinors inducing the 3-Sasakian structure

Abstract

We give a spinorial construction of Sasakian and 3-Sasakian structures in arbitrary dimension, generalizing previously known results in dimensions 5 and 7. Furthermore, we obtain a complete description of the space of invariant spinors on a homogeneous 3-Sasakian space, and show that it is spanned by the Clifford products of invariant differential forms with a certain invariant Killing spinor. Finally, we give a basis for the space of invariant Riemannian Killing spinors on a homogeneous 3-Sasakian space, and determine which of these induce the homogeneous 3-Sasakian structure.