Generalized Killing spinors on spheres

Abstract

We study generalized Killing spinors on round spheres ${\mathbb{S}}^n$. We show that on the standard sphere ${\mathbb{S}}^8$ any generalized Killing spinor has to be an ordinary Killing spinor. Moreover we classify generalized Killing spinors on ${\mathbb{S}}^n$ whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola--Friedrich's canonical spinor on 3-Sasakian manifolds of dimension 7. Finally we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors.

16 pages; new version filling a gap in the proof of Lemma 4.1 which was brought to our attention by Stanislav Wiechmann

References in corpus (4)

• Half-flat structures on S^3xS^3
• The Cauchy problems for Einstein metrics and parallel spinors
• Generalized Killing spinors on Einstein manifolds
• The energy-momentum tensor as a second fundamental form

Cited by in corpus (9)

• Generalized Killing spinors and Lagrangian graphs
• Cauchy problems for Lorentzian manifolds with special holonomy
• Invariant Spinors on Homogeneous Spheres
• Quaternionic Heisenberg groups as naturally reductive homogeneous spaces
• Spinors of real type as polyforms and the generalized Killing equation
• Cauchy spinors on $3$-manifolds
• Generalised Killing Spinors on Three-Dimensional Lie Groups
• Homogeneous Sasakian and 3-Sasakian Structures from the Spinorial Viewpoint
• Parallel spinors for ${\mathrm{G}}_2^{\ast }$ and isotropic structures