Isometric immersions of hypersurfaces in 4-dimensional manifolds via spinors

We give a spinorial characterization of isometrically immersed hypersurfaces into 4-dimensional space forms and product spaces M3(κ)×RM3(κ)×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 on the spinorial Weierstraß representation by U. Abresch, D. Sullivan, R. Kusner, N. Schmidt and many others. 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 some non-existence results for isometric immersions into the 4-dimensional Euclidean space.