Skip to main content

Basic Morse–Novikov cohomology for foliations

Abstract

In this paper we find sufficient conditions for the vanishing of the Morse–Novikov cohomology on Riemannian foliations. We work out a Bochner technique for twisted cohomological complexes, obtaining corresponding vanishing results. Also, we generalize for our setting vanishing results from the case of closed Riemannian manifolds. Several examples are presented, along with applications in the context of l.c.s. and l.c.K. foliations.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Álvarez López, J.A.: The basic component of the mean curvature of Riemannian foliations. Ann. Glob. Anal. Geom. 10, 179-194 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bădiţoiu, G., Escobales, R., Ianuş, S.: A cohomology $(p+1)$ form canonically associated with certain codimension-q foliations on a Riemannian manifold. Tokyo J. Math. 29, 247-270 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Barletta, E., Dragomir, S.: On transversally holomorphic maps of Kählerian foliations. Acta Appl. Math. 54, 121-134 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Dragomir, S., Ornea, L.: Locally Conformal Kähler Geometry, Progr. in Math. 155. Birkhäuser, Boston (1998)

    Book  MATH  Google Scholar 

  5. 5.

    Carrière, Y.: Flots riemanniens. Astérisque 116, 31-52 (1984)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Connes, A.: A survey on foliations and operator algebra and applications. Proc. Symp. Pure Math. 38(I), 521-628 (1982)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Domínguez, D.: Finiteness and tenseness theorems for Riemannian foliations. Am. J. Math. 120, 1237-1276 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Gallot, S., Meyer, D.: Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne. J. Math. Pures Appl. 54, 259-284 (1975)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Gilkey, P.: Index theory, the heat equation, and the Atiyah-Singer index theorem. CRC Press, Boca Raton (1995)

    MATH  Google Scholar 

  10. 10.

    Guedira, F., Lichnerowicz, A.: Géometrie des algèbres de Lie locales de Kirillov. J. Math. Pures Appl. 63, 407-484 (1984)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Habib, G., Richardson, K.: Modified differentials and basic cohomology. J. Geom. Anal. 23, 1314-1342 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Hebda, J.: Curvature and focal points in Riemannian foliations. Indiana Univ. Math. J. 35, 321-331 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Ida, C., Popescu, P.: On the stability of transverse locally conformally symplectic structures. BSG Proc. 20, 1-8 (2013)

    MathSciNet  Google Scholar 

  14. 14.

    Ionescu, A.M., Slesar, V., Vişinescu, M., Vîlcu, G.E.: Transverse killing and twistor spinors associated to the basic Dirac operators. Rev. Math. Phys. 25, 1330011 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    de León, M., López, B., Marrero, J.C., Padrón, E.: On the computation of the Lichnerowicz-Jacobi cohomology. J. Geom. Phys. 44, 507-522 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Lichnerowicz, A.: Les variétés de Poisson et leurs algèbres de Lie associées. J. Diff. Geom. 12(2), 253-300 (1977)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Milnor, J.: Curvature of left invariant metrics of Lie groups. Adv. Math. 21, 293-329 (1967)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Min-Oo, M., Ruh, E., Tondeur, P.: Vanishing theorems for the basic cohomology of Riemannian foliations. J. Reine Angew. Math. 415, 167-174 (1991)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Molino, P.: Riemannian Foliations. Progress in Math, vol. 73. Birkhauser Verlag, Boston Inc., Boston (1988)

    Book  Google Scholar 

  20. 20.

    O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459-469 (1966)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Ornea, L., Verbitsky, M.: Structure theorem for compact Vaisman manifolds. Math. Res. Lett. 10, 799-805 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Ornea, L., Verbitsky, M.: Morse-Novikov cohomology of locally conformally Kähler manifolds. J. Geom. Phys. 59, 295-305 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Park, E., Richardson, K.: The basic Laplacian of a Riemannian foliation. Am. J. Math. 118, 1249-1275 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Pajitnov, A.V.: An analytic proof of the real part of Novikov’s inequalities. Sov. Math. Dokl. 35(2), 456-457 (1987)

    Google Scholar 

  25. 25.

    Poor, W.A.: Differential Geometric Structures. McGraw-Hill, New York (1981)

    MATH  Google Scholar 

  26. 26.

    Reinhart, B.: Foliated manifolds with bundle-like metrics. Ann. Math. 69, 119-132 (1959)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Tondeur, Ph: Geometry of Foliations. Birkhäuser, Basel (1997)

    Book  MATH  Google Scholar 

  28. 28.

    Vaisman, I.: Generalized Hopf manifolds. Geom. Dedic. 13, 231-255 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Vaisman, I.: Remarkable operators and commutation formula on locally conformal Kähler manifolds. Compos. Math. 40, 227-259 (1980)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Vaisman, I.: Locally conformal symplectic manifolds. Int. J. Math. Math. Sci. 8(3), 521-536 (1985)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors thank the referee for very carefully reading a first version of the paper and for his or her most useful suggestions.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Vladimir Slesar.

Additional information

L.O. is partially supported by CNCS UEFISCDI, Project Number PN-II-ID-PCE-2011-3-0118.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ornea, L., Slesar, V. Basic Morse–Novikov cohomology for foliations. Math. Z. 284, 469–489 (2016). https://doi.org/10.1007/s00209-016-1662-5

Download citation

Keywords

  • Riemannian foliations
  • Morse–Novikov cohomology
  • Locally conformally symplectic manifolds

Mathematics Subject Classification

  • 53C12
  • 58A12
  • 53C21