Skip to main content
SpringerLink
Log in

A vanishing result for the Spin c Dirac operator defined along the leaves of a foliation

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

In the setting of a closed Riemannian manifold endowed with a smooth, non-necessarily integrable distribution, we extend a Lichnerowicz type formula which is known to work in the particular case of a transverse bundle associated to a Riemannian foliation. Interesting settings in which non-integrable distributions appear naturally are emphasized. As an application, we consider the distribution as being even dimensional and integrable; we consider also a hermitian line bundle, with a hermitian connection, such that the induced curvature tensor is non-degenerate, and an arbitrary hermitian bundle endowed also with a hermitian connection. Taking the k power of the line bundle and canonically constructing a Spin c Dirac operator defined along the leaves of the foliation generated by the distribution, we prove a vanishing result for the half kernel of this operator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Álvarez López J.A., Kordyukov Y.A.: Adiabatic limits and spectral sequences for Riemannian foliations. Geom. Funct. Anal. 10, 977–1027 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berline N., Getzler E., Vergne M.: Heat Kernels and Dirac Operators. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  3. Braverman, M.: Vanishing theorems for the half-kernel of a Dirac operator, arXiv:math/9805127 v2[math.DG]. http://arxiv.org/abs/math/9805127 (1998)

  4. Craioveanu M., Puta M.: Asymptotic properties of eigenvalues of the basic Laplacian associated to certain Riemannian foliations. Bull. Math. Soc. Sci. Math. Roumanie 35, 61–65 (1991)

    MathSciNet  Google Scholar 

  5. Craioveanu M., Slesar V.: A Weitzenbock formula for a closed Riemannian manifolds with two orthogonal complementary distributions. Bull. Math. Soc. Sci. Math. Roumanie 52(3), 271–279 (2009)

    MathSciNet  Google Scholar 

  6. Connes A., Skandalis G.: The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. Kyoto 20, 1139–1183 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  7. Douglas R.G., Glazebrook J.F., Kamber F.W., Yu G.: Index formulas for geometric Dirac operators in Riemannian foliations. K-Theory 9, 407–441 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Glazebruck J.F., Kamber F.W.: Transversal Dirac families in Riemannain foliations. Commun. Math. Phys. 140, 217–240 (1991)

    Article  Google Scholar 

  9. Heitsch J., Lazarov C.: A Lefschetz theorem for foliated manifolds. Topology 29, 127–162 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jung S.D.: The first eigenvalue of the transversal Dirac operator. J. Geom. Phys. 39, 253–264 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kamber F.W., Tondeur Ph.: De Rham-Hodge theory for Riemannian foliations. Math. Ann. 277, 425–431 (1987)

    Article  MathSciNet  Google Scholar 

  12. Kordyukov Y.A.: Vanishing theorem for transverse Dirac operators on Riemannian foliations. Ann. Glob. Anal. Geom. 34, 195–211 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ma X., Marinescu G.: The Spin c Dirac operator on high tensor powers of a line bundle. Math. Z 240, 651–664 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ma X., Marinescu G.: Holomorphic Morse Inequalities and Bergman Kernels Progress in Math 254. Birkhäuser, Basel (2007)

    Google Scholar 

  15. McDuff D., Salamon D.: Introduction to Symplectic Topology. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  17. Nicolaescu L.: Geometric connections and geometric Dirac operators on contact manifolds. Diff. Geom. Appl. 22, 355–378 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Ponge R.: Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry. J. Funct. Anal. 252, 399–463 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Prokhorenkov Y., Richardson K.: Natural equivariant transversally elliptic Dirac operators. Geom. Dedic. 151, 411–429 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Roe J.: Finite propogation speed and Connes’ foliation algebra. Math. Proc. Camb. Phil. Soc. 102, 459–466 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. Slesar V.: Weitzenböck formulas for Riemannian foliations. Diff. Geom. Appl. 27, 362–367 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  26. Vacaru S.: Spinor structures and nonlinear connections in vector bundles, generalized Lagrange and Finsler spaces. J. Math. Phys. 37, 508–523 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Sciences, University of Craiova, 13, Al.I. Cuza St., 200585, Craiova, Romania

    Vladimir Slesar

Authors
  1. Vladimir Slesar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir Slesar.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Slesar, V. A vanishing result for the Spin c Dirac operator defined along the leaves of a foliation. Geom Dedicata 161, 239–249 (2012). https://doi.org/10.1007/s10711-012-9704-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-012-9704-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation