Mathematische Zeitschrift

, Volume 249, Issue 3, pp 545-580

First online:

Generalized cylinders in semi-Riemannian and spin geometry

  • Christian BärAffiliated withInstitut für Mathematik, Universität Potsdam Email author 
  • , Paul GauduchonAffiliated withCentre de Mathématiques, École Polytechnique
  • , Andrei MoroianuAffiliated withCentre de Mathématiques, École Polytechnique

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We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics.