Killing spinors and hypersurfaces

TL;DR

This paper explores the geometry of hypersurfaces in Einstein spin manifolds with Killing spinors, establishing PDE characterizations and embedding results for such structures in both Riemannian and indefinite settings.

Contribution

It introduces a PDE framework for hypersurfaces with induced spinors and proves an embedding theorem for real analytic pseudo-Riemannian manifolds with these spinor conditions.

Findings

Characterization of hypersurfaces via a PDE for induced spinors

Embedding theorem for real analytic pseudo-Riemannian manifolds with Killing spinors

Description of intrinsic geometry in Einstein spin manifolds

Abstract

We consider spin manifolds with an Einstein metric, either Riemannian or indefinite, for which there exists a Killing spinor. We describe the intrinsic geometry of nondegenerate hypersurfaces in terms of a PDE satisfied by a pair of induced spinors, akin to the generalized Killing spinor equation. Conversely, we prove an embedding result for real analytic pseudo-Riemannian manifolds carrying a pair of spinors satisfying this condition.