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TRANSVERSE KILLING FORMS ON COMPLETE FOLIATED RIEMANNIAN MANIFOLDS

  • Jung, Seoung Dal (Department of Mathematics and Research Institute for Basic Sciences, Jeju National University)
  • Received : 2014.08.04
  • Accepted : 2014.09.11
  • Published : 2014.12.25

Abstract

In this article, we study the transverse Killing forms with finite global norms on complete foliated Riemannian manifolds.

Keywords

References

  1. J. A. Alvarez Lopez, The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom. 10 (1992), 179-194. https://doi.org/10.1007/BF00130919
  2. T. Aoki and S. Yorozu, $L^2$-transverse conformal and Killing fields on complete foliated Riemannian manifolds, Yokohama Math. J. 36 (1988), 27-41.
  3. P. Berard, A note on Bochner type theorems for complete manifolds, Manuscripta Math. 69 (1990), 261-266. https://doi.org/10.1007/BF02567924
  4. S. D. Jung, The first eigenvalue of the transversal Dirac operator, J. Geom. Phys. 39 (2001), 253-264. https://doi.org/10.1016/S0393-0440(01)00014-6
  5. S. D. Jung, Eigenvalue estimates for the basic Dirac operator on a Riemannian foliation admitting a basic harmonic 1-form, J. Geom. Phys. 57 (2007), 1239- 1246. https://doi.org/10.1016/j.geomphys.2006.04.008
  6. M. J. Jung and S. D. Jung, Liouville type theorem for transversally harmonic maps, arXiv:1307.3627v2[math.DG] 29 Aug 2014.
  7. S. D. Jung and M. J. Jung, Transverse Killing forms on a Kahler foliation, Bull. Korean Math. Soc. 49 (2012), 445-454. https://doi.org/10.4134/BKMS.2012.49.3.445
  8. S. D. Jung and K. Richardson, Transverse conformal Killing forms and a Gallot-Meyer theorem for foliations, Math. Z. 270 (2012), 337-350. https://doi.org/10.1007/s00209-010-0800-8
  9. F. W. Kamber and Ph. Tondeur, Infinitesimal automorphisms and second variation of the energy for harmonic foliations, Tohoku Math. J. 34 (1982), 525-538. https://doi.org/10.2748/tmj/1178229154
  10. F. W. Kamber and Ph. Tondeur, De Rham-Hodge theory for Riemannian foliations, Math. Ann. 277 (1987), 415-431. https://doi.org/10.1007/BF01458323
  11. T. Kashiwada, On conformal Killing tensor, Natur. Sci. Rep. Ochanomizu Univ. 19 (1968), 67-74.
  12. T. Kashiwada and S. Tachibana, On the integrability of Killing-Yano's equation, J. Math. Soc. Japan 21 (1969), 259-265. https://doi.org/10.2969/jmsj/02120259
  13. P. Molino, Riemannian foliations, translated from the French by Grant Cairns, Boston: Birkhaser, 1988.
  14. S. Nishikawa and Ph. Tondeur, Transversal infinitesimal automorphisms of harmonic foliations on complete manifolds, Anal. Global Anal. Geom. 7 (1989), 47-57. https://doi.org/10.1007/BF00137401
  15. J. S. Pak and S. D. Jung, A transversal Dirac operator and some vainshing theorems on a complete foliated Riemannian manifold, Math. J. Toyama Univ. 16 (1993), 97-108.
  16. J. S. Pak and S. Yorozu, Transverse fields on foliated Riemannian manifolds, J. Korean Math. Soc. 25 (1988), 83-92.
  17. E. Park and K. Richardson, The basic Laplacian of a Riemannian foliation, Amer. J. Math. 118 (1996), 1249-1275. https://doi.org/10.1353/ajm.1996.0053
  18. Ph. Tondeur, Geometry of foliations, Basel: Birkhauser Verlag, 1997.
  19. K. Yano, Some remarks on tensor fields and curvature, Ann. Math. 55 (1952), 328-347. https://doi.org/10.2307/1969782
  20. S. Yorozu, Conformal and Killing vector fields on complete non-compact Riemannian manifolds, Geometry of geodesics and related topics, Advanced Studies in Pure Mathematics 3 (1984).
  21. S. Yorozu, Killing vector fields on complete Riemannian manifolds, Proc. Amer. Math. Soc. 84 (1982), 115-120.
  22. S. Yorozu, The non-existence of Killing fields, Tohoku Math. J. 36 (1984), 99-105. https://doi.org/10.2748/tmj/1178228906

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  1. $$L^\mathrm{2}$$ L 2 -transverse conformal Killing forms on complete foliated manifolds 2018, https://doi.org/10.1007/s00209-017-1905-0