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Stability of the Reeb vector field of H-contact manifolds

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Abstract

It is well known that a Hopf vector field on the unit sphere S 2n+1 is the Reeb vector field of a natural Sasakian structure on S 2n+1. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction.

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References

  1. Blair D.E.: Riemannian geometry of contact and symplectic manifold. Progress in Math. 203. Birkhäuser, Boston (2002)

    Google Scholar 

  2. Blair D.E., Koufogiorgos T., Papantoniou B.J.: Contact metric manifolds satisfying a nullity condition. Israel J. Math. 91, 189–214 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Boeckx E., González-Dávila J.C., Vanhecke L.: Stability of the geodesic flow for the energy. Comm. Math. Univ. Carolinae 43, 201–213 (2002)

    MATH  Google Scholar 

  4. Boothby W.M.: An introduction to differentiable manifolds and Riemannian geometry. Academic Press, New York (1975)

    MATH  Google Scholar 

  5. Boyer C., Galicki K.: Einstein manifolds and contact geometry. Proc. Amer. Math. Soc. 129, 2419–2430 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  6. Boyer C., Galicki K.: On Sasakian-Einstein geometry. Intern. J. Math. 11, 873–909 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Borrelli V.: Stability of the Reeb vector field of a Sasakian manifold. Soochow J. Math. 30(3), 283–292 (2004)

    MATH  MathSciNet  Google Scholar 

  8. Borrelli V., Gil-Medrano O.: A critical radius for unit Hopf vector fields on spheres. Math. Ann. 324(4), 731–751 (2006)

    Article  MathSciNet  Google Scholar 

  9. Brito F.: Total bending of flows with mean curvature correction. Differ. Geom. Appl. 12, 157–163 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chern S.S., Hamilton R.S.: On Riemannian metrics adapted to three-dimensional contact manifolds. Lecture Notes in Math. 1111, pp. 279–305. Springer-Verlag, Berlin (1985)

    Google Scholar 

  11. Gil-Medrano O.: Relationiship between volume and energy of unit vector fields. Differ. Geom. Appl. 15, 137–152 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gil-Medrano, O.: Unit vector fields that are critical points of the volume and of the energy: cha-racterization and examples. Proceedings of the International Conference “Curvature in Geometry” in honour of Prof. L. Vanhecke, Progress in Math. 234, Birkhäuser, Boston, pp. 165–186 (2005)

  13. Gil-Medrano O., Llinares-Fuster E.: Second variation of volume and energy of vector fields. Stability of Hopf vector fields. Math. Ann. 320, 531–545 (2001)

    MATH  MathSciNet  Google Scholar 

  14. Gil-Medrano O., Llinares-Fuster E.: Minimal unit vector fields. Tohoku Math. J. 54, 71–84 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gluck H., Ziller W.: On the volume of a unit vector field on the three sphere. Comment. Math. Helv. 61, 177–192 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  16. González-Dávila J.C., Vanhecke L.: Examples of minimal unit vector fields. Ann. Global Anal. Geom. 18, 385–404 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Koufogiorgos T., Tsichlias C.: On the existence of a new class of contact metric manifolds. Canad. Math. Bull. 43, 440–447 (2000)

    MATH  MathSciNet  Google Scholar 

  18. Milnor J.: Curvature of left invariant metrics on Lie groups. Adv. in Math. 21, 293–329 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  19. Perrone D.: Contact Riemannian manifolds satisfying R(X, ξ)R = 0. Yokohama Math. J. 39, 141–149 (1992)

    MATH  MathSciNet  Google Scholar 

  20. Perrone D.: Ricci tensor and spectral rigidity of contact Riemannian three-manifolds. Bull. Inst. Math. Acad. Sin. 24, 127–138 (1996)

    MATH  MathSciNet  Google Scholar 

  21. Perrone D.: Homogeneous contact Riemannian three-manifolds. Ill. Math. J. 42, 243–256 (1998)

    MATH  MathSciNet  Google Scholar 

  22. Perrone D.: Harmonic characteristic vector fields on contact metric three-manifolds. Bull. Aust. Math. Soc 67, 305–315 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Perrone D.: Contact metric manifolds whose characteristic vector field is a harmonic vector field. Differ. Geom. Appl. 20, 367–378 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Perrone D.: The rough Laplacian and harmonicity of Hopf vector fields. Ann. Global Anal. Geom. 28, 91–106 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Poor W.A.: Differential geometric structures. McGraw Hill, New York (1981)

    MATH  Google Scholar 

  26. Tanno S.: The topology of contact Riemannian manifolds. Ill. Math. J. 12, 700–717 (1968)

    MATH  MathSciNet  Google Scholar 

  27. Tanno S.: Variational problems on contact Riemannian manifolds. Trans. Am. Math. Soc. 314, 349–379 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  28. Watanabe Y.: Geodesic symmetries in Sasakian locally ϕ-symmetric spaces. Kodai Math. J. 3(1), 48–55 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  29. Wiegmink G.: Total bending of vector fields on Riemannian manifolds. Math. Ann. 303, 325–344 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  30. Wiegmink G.: Total bending of vector fields on the sphere S 3. Differ. Geom. Appl. 6, 219–236 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  31. Wood C.M.: On the energy of a unit vector field. Geom. Dedicata 64, 319–330 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  32. Wood C.M.: The energy of Hops vector field. Manuscr. Math. 101, 71–78 (2000)

    Article  MATH  Google Scholar 

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  1. Dipartimento di Matematica “Ennio De Giorgi”, Università del Salento, 73100, Lecce, Italy

    Domenico Perrone

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  1. Domenico Perrone
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Correspondence to Domenico Perrone.

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Supported by funds of the University of Lecce and M.I.U.R.(PRIN).

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Perrone, D. Stability of the Reeb vector field of H-contact manifolds. Math. Z. 263, 125–147 (2009). https://doi.org/10.1007/s00209-008-0413-7

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