Isometric Immersions of Hypersurfaces in 4-dimensional Manifolds via Spinors

Marie-Amélie Lawn (1) , Julien Roth (2)

1 Unité de recherche en mathématiques
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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.

Mots clés: Dirac Operator; Generalized Killing Spinors; Isometric Immersions; Gauss and Codazzi-Mainardi Equations; Energy-Momentum Tensor

Domaines: Géométrie différentielle [math.DG]

Marie-Amélie Lawn, Julien Roth. Isometric Immersions of Hypersurfaces in 4-dimensional Manifolds via Spinors. Differential Geometry and its Applications, 2010, 28 (2), pp.205-219. ⟨hal-00264969v2⟩