Mathematische Zeitschrift

, Volume 249, Issue 3, pp 545–580 | Cite as

Generalized cylinders in semi-Riemannian and spin geometry

  • Christian BärEmail author
  • Paul Gauduchon
  • Andrei Moroianu


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.


Dirac Operator Constant Curvature Fundamental Theorem Natural Manner Variation Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bär, C.: Real Killing spinors and holonomy. Commun. Math. Phys. 154, 509–521 (1993)Google Scholar
  2. 2.
    Baum, H.: Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten. B. G. Teubner Verlagsgesellschaft, Leipzig, 1981Google Scholar
  3. 3.
    Bourguignon, J.-P., Gauduchon, P.: Spineurs, opérateurs de Dirac et variations de métriques. Commun. Math. Phys. 144, 581–599 (1992)zbMATHGoogle Scholar
  4. 4.
    Deligne, P. et al., (eds.), Quantum fields and strings: A course for mathematicians, Vol. 1, AMS, 1999Google Scholar
  5. 5.
    Friedrich, T., Kim, E. C.: Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors. J. Geom. Phys. 37, 1–14 (2001)CrossRefGoogle Scholar
  6. 6.
    Kobayashi, S., Nomizu, K.: Foundations of differential geometry, Vol. 2, Interscience Publishers, John Wiley & Sons, New York, Chichester, Brisbane, Toronto, 1969Google Scholar
  7. 7.
    Morel, B.: The energy-momentum tensor as a second fundamental form. Preprint, 2003, math.DG/0302205Google Scholar
  8. 8.
    Mounoud, P.: Some topological and metrical properties of the space of Lorentz metrics. Diff. Geom. Appl. 15, 47–57 (2001)Google Scholar
  9. 9.
    O’Neill, B.: Semi-Riemannian geometry. Academic Press, New York, London, 1983Google Scholar
  10. 10.
    Petersen, P.: Riemannian geometry. Springer Verlag, New York, 1998Google Scholar
  11. 11.
    Wolf, J.: Spaces of constant curvature. Publish or Perish, Wilmington, Delaware, 1984Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Christian Bär
    • 1
    Email author
  • Paul Gauduchon
    • 2
  • Andrei Moroianu
    • 2
  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany
  2. 2.Centre de MathématiquesÉcole PolytechniquePalaiseau CedexFrance

Personalised recommendations