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Homogeneous Sasakian and 3-Sasakian Structures from the Spinorial Viewpoint

Jordan Hofmann

TL;DR

The paper develops a uniform spinorial framework to realize Sasakian and 3-Sasakian geometries from Killing spinors in arbitrary dimension, extending known results in dimensions 5 and 7. It provides a complete description of invariant spinors on simply-connected homogeneous 3-Sasakian spaces, showing they form an algebra isomorphic to invariant $\varphi_1$-anti-holomorphic differential forms, generated by $\omega$ and $y_1$, and gives an explicit basis for invariant Killing spinors. The work proves that invariant Killing spinors induce invariant Sasakian/3-Sasakian structures on homogeneous spaces and identifies which spinors recover the homogeneous structure, with a detailed construction of $\psi_k = \omega^{k+1} - i(k+1) y_1 \wedge \omega^k$ as a canonical basis. By combining spinorial methods, invariant theory, and Nomizu data, the paper extends the spinorial correspondence between Einstein-Sasakian and 3-Sasakian structures to a broad class of homogeneous spaces, providing concrete algebraic and geometric classifications that deepen the link between spin geometry and quaternionic contact structures.

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.

Homogeneous Sasakian and 3-Sasakian Structures from the Spinorial Viewpoint

TL;DR

The paper develops a uniform spinorial framework to realize Sasakian and 3-Sasakian geometries from Killing spinors in arbitrary dimension, extending known results in dimensions 5 and 7. It provides a complete description of invariant spinors on simply-connected homogeneous 3-Sasakian spaces, showing they form an algebra isomorphic to invariant -anti-holomorphic differential forms, generated by and , and gives an explicit basis for invariant Killing spinors. The work proves that invariant Killing spinors induce invariant Sasakian/3-Sasakian structures on homogeneous spaces and identifies which spinors recover the homogeneous structure, with a detailed construction of as a canonical basis. By combining spinorial methods, invariant theory, and Nomizu data, the paper extends the spinorial correspondence between Einstein-Sasakian and 3-Sasakian structures to a broad class of homogeneous spaces, providing concrete algebraic and geometric classifications that deepen the link between spin geometry and quaternionic contact structures.

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.
Paper Structure (10 sections, 37 theorems, 88 equations, 1 figure, 2 tables)

This paper contains 10 sections, 37 theorems, 88 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Spin Representation via Exterior Forms
  4. Spinors on Homogeneous Spaces
  5. Sasakian and ${\text{3-Sasakian}}$ Structures
  6. Homogeneous ${\text{3-Sasakian}}$ Spaces
  7. The ($3$-)Sasakian Structures Induced by Killing Spinors
  8. Invariance of Spinors and their Associated ($3$-)Sasakian Structures
  9. Invariant Differential Forms and Spinors
  10. The Space of Riemannian Killing Spinors

Key Result

Theorem 1

Let $(M,g)$ be a Riemannian spin manifold carrying a pair $\psi_1,\psi_2$ of Killing spinors (resp. four Killing spinors $\psi_1,\psi_2,\psi_3,\psi_4$) for the same Killing number $\lambda \in \{\frac{1}{2},\frac{-1}{2}\}$. If the vector field $\xi_{\psi_1,\psi_2}$ defined by the equation for all $X\in TM$ has locally constant non-zero length (resp. if the vector fields $\xi_{\psi_1,\psi_2}$, $\x

Figures (1)

  • Figure 1: Homogeneous Spin Structure

Theorems & Definitions (74)

  • Theorem
  • Theorem
  • Theorem
  • Remark 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • ...and 64 more