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Journal of Geometry

, Volume 58, Issue 1–2, pp 66–86 | Cite as

Curvature of indefinite almost contact manifolds

  • Agustín Bonome
  • Regina Castro
  • Eduardo García-Río
  • Luis Hervella
Article

Abstract

The curvature tensor of indefinite almost contact manifolds is investigated. By means of the study of the Jacobi operator along spacelike, timelike and null geodesies, spaces of constant curvature are characterized as well as spaces of pointwise constant ϕ-sectional curvature. As an extension of these conditions we introduce the socalled ϕ-isotropic spaces and show a local classification of such manifolds.

Keywords

Curvature Tensor Constant Curvature Jacobi Operator Contact Manifold Local Classification 
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.

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References

  1. [1]
    BARROS, M. and ROMERO, A., Indefinite Kähler manifolds, Math. Ann.,261 (1982), 55–62.Google Scholar
  2. [2]
    BONOME, A., CASTRO, R., GARCIA-RIO, E. and HERVELLA, L.M., On the holomorphic sectional curvature of an indefinite Kähler manifold, C. R. Acad. Sci. Paris,315 (1992), 1183–1187.Google Scholar
  3. [3]
    BONOME, A., CASTRO, R., GARCIA-RIO, E., HERVELLA, L.M. and MATSUSHITA, Y., Null holomorphically flat indefinite almost Hermitian manifolds, Illinois J. Math.,39 (1995), to appear.Google Scholar
  4. [4]
    CENDAN-VERDES, J., GARCIA-RIO, E. and VAZQUEZ-ABAL, M.E., On the semi-Riemannian structure of the tangent bundle of a two-point homogeneous space, Riv. Mat. Univ. Parma, (5)3 (1994), 253–270.Google Scholar
  5. [5]
    DAJCZER, M. and NOMIZU, K., On sectional curvature of indefinite metrics II, Math. Ann.,247 (1980), 279–282.Google Scholar
  6. [6]
    DAJCZER, M. and NOMIZU, K., On the boundness of Ricci curvature of indefinite metrics, Bol. Soc. Brasil Math.,11 (1980), 25–30.Google Scholar
  7. [7]
    GRAVES, L. and NOMIZU, K., On sectional curvature of indefinite metrics, Math. Ann.,232 (1978), 267–272.Google Scholar
  8. [8]
    JANSSENS, D. and VANHECKE, L., Almost contact structures and curvature tensors, Kōdai Math. J.,4 (1981), 1–27.Google Scholar
  9. [9]
    KENMOTSU, K., A class of almost contact Riemannian manifolds, Tôhoku Math. J.,24 (1972), 93–103.Google Scholar
  10. [10]
    KULKARNI, R.S., The values of sectional curvatures in indefinite metrics, Comment. Math. Helv.,54 (1979), 173–176.Google Scholar
  11. [11]
    NOMIZU, K., Remarks on sectional curvature of an indefinite metric, Proc. Amer. Math. Soc.,89 (1983), 473–476.Google Scholar
  12. [12]
    OLSZAK, Z., On the existence of generalized complex space forms, Israel J. Math.,65 (1989), 214–218.Google Scholar
  13. [13]
    O'NEILL, B., Semi-Riemannian Geometry with applications to Relativity, Academic Press, New-York, 1983.Google Scholar
  14. [14]
    TAKAHASHI, T., Sasakian manifolds with pseudo-Riemannian metric, Tôhoku Math. J.,21 (1969), 271–290.Google Scholar
  15. [15]
    TANNO, S., Constancy of holomorphic sectional curvature in almost Hermitian manifolds, Kōdai Math. Sem. Rep.,25 (1973), 190–201.Google Scholar
  16. [16]
    TONDEUR, PH., Foliations on Riemannian manifolds, Universitext, Springer-Verlag, Berlin, Heidelberg, New York, 1988.Google Scholar
  17. [17]
    TRICERRI, F. and VANHECKE, L., Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc.,267 (1981), 365–398.Google Scholar
  18. [18]
    YANO, K. and ISHIHARA, S., Tangent and cotangent bundles, Pure and Applied Math. 16, Marcel Dekker, New York, 1973.Google Scholar
  19. [19]
    YANO, K. and KON, M., Structures on Manifolds, Series in Pure Mathematics 3, World Scientific Publ. Co., Singapore, 1984.Google Scholar

Copyright information

© Birkhäuser Verlag 1997

Authors and Affiliations

  • Agustín Bonome
    • 1
  • Regina Castro
    • 1
  • Eduardo García-Río
    • 2
  • Luis Hervella
    • 1
  1. 1.Departamento de Xeometría e TopoloxíaUniversidade de Santiago de CompostelaSantiago de CompostelaSpain
  2. 2.Departamento de Análise Matemática Facultade de MatemáticasUniversidade de Santiago de CompostelaSantiago de CompostelaSpain

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