Isometric Immersions of Hypersurfaces in 4-dimensional Manifolds via Spinors

Details Isometric Immersions of Hypersurfaces in 4-dimensional Manifolds via Spinors Isometric Immersions of Hypersurfaces in 4-dimensional Manifolds via Spinors

2008-03-18

arxiv.org/abs/0803.2621v2

Differential Geometry and its Applications 28, 2 (2010) 1045-1061

Mathematics

Differential Geometry

21 pages, application added

Scientific paper

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.

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