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Rigidity results for spin manifolds with foliated boundary

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Abstract

In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O’Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold. As a consequence, we show that the flow is a local product. In particular, in the case where solutions of the basic Dirac equation are given by basic Killing spinors, we characterize the geometry of the manifold and the flow.

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References

  1. Alexandrov B., Grantcharov G., Ivanov S.: An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting a parallel one-form. J. Geom. Phys. 28(3–4), 263–270 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bär C.: Real Killing spinors and holonomy. Commun. Math. Phys. 154, 509–521 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bär C.: The Dirac operator on space forms of positive curvature. J. Math. Soc. Jpn. 48, 69–83 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bär C.: On nodal sets for Dirac and Laplace operators. Commun. Math. Phys. 188, 709–721 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bär C.: Extrinsic bounds for eigenvalues of the Dirac operator. Ann. Glob. Anal. Geom. 16, 573–596 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bourguignon, J.-P., Hijazi, O., Milhorat, J.-L., Moroianu, A., Moroianu, S.: A spinorial approach to Riemannian and conformal geometry. In: EMS Monographs in Mathematics (to appear)

  7. Chen B.-Y.: Riemannian submersions, minimal immersions and cohomology class. Proc. Jpn. Acad. 81, 162–167 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. El Kacimi A., Gmira B.: Stabilité du caractère kählérien transverse. Isr. J. Math. 101, 323–347 (1997)

    Article  Google Scholar 

  9. El Kacimi A.: Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications. Compositio Mathematica 73, 57–106 (1990)

    MathSciNet  MATH  Google Scholar 

  10. Friedrich, T.: Dirac operators in Riemannian geometry. In: Graduate Studies in Mathematics, vol. 25. American Mathematical Society, Providence (2000)

  11. Ginoux, N.: The Dirac spectrum. In: Lecture Notes in Mathematics, vol. 1976. Springer, Berlin (2009)

  12. Glazebrook J.F., Kamber F.W.: On spectral flow of transversal Dirac operators and a theorem of Vafa–Witten. Ann. Glob. Anal. Geom. 9, 27–35 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Glazebrook J.F., Kamber F.W.: Transversal Dirac families in Riemannian foliations. Commun. Math. Phys. 140, 217–240 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Habib, G.: Tenseur d’impulsion-énergie et feuilletages. Ph.D. thesis, Université Henri Poincaré-Nancy I (2006)

  15. Hijazi O., Montiel S.: Extrinsic Killing spinors. Math. Zeit. 244, 337–347 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Hijazi O., Montiel S.: A spinorial characterization of hyperspheres. Calc. Var. Part. Differ. Equ. 48, 527–544 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hijazi O., Montiel S.: A holographic principle for the existence of parallel spinors and an inequality of Shi-Tam type. Asian J. Math. 18(3), 489–506 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hijazi O., Montiel S., Roldan S.: Eigenvalue boundary problems for the Dirac operator. Commun. Math. Phys. 231, 375–390 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hijazi O., Montiel S., Zhang X.: Dirac operator on embedded hypersurfaces. Math. Res. Lett. 8, 195–208 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton University Press, Princeton (1989)

  21. Li M.: A sharp comparison theorem for compact manifolds with mean convex boundary. J. Geom. Anal. 24, 1490–1496 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Raulot S.: Rigidity of compact Riemannian spin manifolds with boundary. Lett. Math. Phys. 86, 177–192 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Raulot S., Savo A.: On the first eigenvalue of the Dirichlet-to-Neumann operator on forms. J. Funct. Anal. 262, 889–914 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Raulot S., Savo A.: A Reilly formula and eigenvalue estimates for differential forms. J. Geom. Anal. 21, 620–640 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Schwarz, G.: Hodge decomposition—a method for solving boundary value problems. In: Lecture Notes in Mathematics, vol. 1607. Springer, Berlin (1995)

  28. Tondeur, P.: Foliations on Riemannian manifolds. In: Universitext, vol. 140. Springer, Berlin (1988)

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences II, Lebanese University, P.O. Box 90656, Fanar, Matn, Lebanon

    Fida El Chami & Georges Habib

  2. Département informatique, IUT de Metz, Ile du Saulcy, CS 10628, 57045, Metz Cedex 01, France

    Nicolas Ginoux

  3. Department of Mathematics and Statistics, Faculty of Natural and Applied Sciences, Notre Dame University-Louaizé, P.O. Box 72, Zouk Mikael, Lebanon

    Roger Nakad

Authors
  1. Fida El Chami
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  2. Nicolas Ginoux
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  3. Georges Habib
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  4. Roger Nakad
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Correspondence to Roger Nakad.

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El Chami, F., Ginoux, N., Habib, G. et al. Rigidity results for spin manifolds with foliated boundary. J. Geom. 107, 533–555 (2016). https://doi.org/10.1007/s00022-015-0286-y

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  • DOI: https://doi.org/10.1007/s00022-015-0286-y

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