Isometric Immersions of Hypersurfaces in 4-dimensional Manifolds via Spinors

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.
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Submitted on : Tuesday, December 9, 2008 - 3:49:26 PM
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  • HAL Id : hal-00264969, version 2
  • ARXIV : 0803.2621

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

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