Isometric Immersions of Hypersurfaces in 4-dimensional Manifolds via Spinors

TL;DR

This paper characterizes isometric hypersurfaces in 4D space forms and product spaces using special spinor fields, extending previous surface results to higher dimensions and providing a non-existence theorem for hypersurfaces in Euclidean 4-space.

Contribution

It introduces a spinorial characterization of hypersurfaces in 4D manifolds, generalizing recent surface results to higher dimensions with new non-existence findings.

Findings

Characterization of hypersurfaces via generalized Killing spinors

Extension of surface results to higher dimensions

Non-existence of certain hypersurfaces in Euclidean 4-space

Abstract

We give a spinorial characterization of isometrically immersed hypersurfaces into 4-dimensional space forms and product spaces $\M^3(\kappa)\times\R$, in terms of the existence of particular spinor fields, called generalized Killing spinors or equivalently solutions of a Dirac equation. This generalizes to higher dimensions several recent results for surfaces by T. Friedrich, B. Morel and the two authors. The main argument is the interpretation of the energy-momentum tensor of a generalized Killing spinor as the second fundamental form, possibly up to a tensor depending on the ambient space. As an application, we deduce a non-existence result for hypersurfaces in the 4-dimensional Euclidean space.