Annals of Global Analysis and Geometry

, Volume 22, Issue 3, pp 291–300 | Cite as

Pin c and Lipschitz Structures on Products of Manifolds

  • Marcin Bobieński
  • Andrzej Trautman


The topological condition for the existence of a pin c structure on the product of two Riemannian manifoldsis derived and applied to construct examples of manifolds havingthe weaker Lipschitz structure, but no pin c structure.An example of a five-dimensional manifold with this property is given;it is pointed out that there are no manifolds of lower dimension withthis property.

spin pinc and Lipschitz structures topological obstructions structures on products 


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  1. 1.
    Atiyah, M. F., Bott, R. and Shapiro, A.: Clifford modules, Topology 3, Suppl. 1 (1964), 3-38.Google Scholar
  2. 2.
    Cahen, M., Gutt, S. and Trautman, A.: Pin structures and the modified Dirac operator, J. Geom. Phys. 17 (1995), 283-297.Google Scholar
  3. 3.
    D?browski, L. and Trautman, A.: Spinor structures on spheres and projective spaces, J. Math. Phys. 27 (1986), 2022-2028.Google Scholar
  4. 4.
    Friedrich, Th.: Dirac Operators in Riemannian Geometry, Grad. Studies in Math. 25, Amer. Math. Soc., Providence, RI, 2000.Google Scholar
  5. 5.
    Friedrich, Th. and Trautman, A.: Spin spaces, Lipschitz groups, and spinor bundles, Ann. Glob. Anal. Geom. 18 (2000), 221-240.Google Scholar
  6. 6.
    Greenberg, M. J. and Harper, J. R.: Algebraic Topology: A First Course, Math. Lecture Note Series, Benjamin, Reading, MA, 1981.Google Scholar
  7. 7.
    Kirby, R. C. and Taylor, L. R.: Pin structures on low-dimensional manifolds, in: Geometry of Low-Dimensional Manifolds, Vol. 2, London Math. Soc. Lecture Note Series 151, Cambridge Univ. Press, Cambridge, 1990, pp. 177-242.Google Scholar
  8. 8.
    Lawson, Jr., H. B. and Michelsohn, M. L.: Spin Geometry, Princeton Univ. Press, Princeton, NJ, 1989.Google Scholar
  9. 9.
    Milnor, J. W. and Stasheff, J. D.: Characteristic Classes, Ann. of Math. Stud. 76, Princeton Univ. Press, Princeton, NJ, 1974.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Marcin Bobieński
    • 1
  • Andrzej Trautman
    • 2
  1. 1.Department of Mathematical Methods in PhysicsWarsaw UniversityWarszawaPoland
  2. 2.Institute of Theoretical PhysicsWarsaw UniversityWarszawaPoland

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