Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 665–677 | Cite as

\(L^\mathrm{2}\)-transverse conformal Killing forms on complete foliated manifolds

  • Seoung Dal Jung
  • Huili Liu


In this article, we study the \(L^2\)-transverse conformal Killing forms on complete foliated Riemannian manifolds and prove some vanishing theorems. Also, we study the same problems on Kähler foliations with a complete bundle-like metric.


Transverse Killing form Transverse conformal Killing form Mean curvature form 

Mathematics Subject Classification

53C12 53C27 57R30 



The authors would like to thank the referee for the valuable suggestions and the comments. The first author was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2015R1A2A2A01003491) and the second author was supported by NSFC (No. 11371080).


  1. 1.
    Alvarez López, J.A.: The basic component of the mean curvature of Riemannian foliations. Ann. Glob. Anal. Geom. 10, 179–194 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aoki, T., Yorozu, S.: \(L^2\)-transverse conformal and Killing fields on complete foliated Riemannian manifolds. Yokohama Math. J. 36, 27–41 (1988)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Jung, S.D.: Eigenvalue estimate for the basic Dirac operator on a Riemannian foliation admitting a basic harmonic 1-form. J. Geom. Phys. 57, 1239–1246 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Jung, S.D.: Transverse Killing forms on complete foliated Riemannian manifolds. Honam Math. J. 36, 731–737 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jung, S.D.: Transverse conformal Killing forms on Kähler foliations. J. Geom. Phys. 90, 29–41 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jung, M.J., Jung, S.D.: Riemannian foliations admitting transversal conformal fields. Geom. Dedicata 133, 155–168 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jung, S.D., Jung, M.J.: Transverse Killing forms on a Kähler foliation. Bull. Korean Math. Soc. 49, 445–454 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jung, S.D., Richardson, K.: Transverse conformal Killing forms and a Gallot–Meyer theorem for foliations. Math. Z. 270, 337–350 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jung, S.D., Jung, M.J.: Transversally holomorphic maps between Kähler foliations. J. Math. Anal. Appl. 416, 683–697 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jung, M.J., Jung, S.D.: Liouville type theorems for transversally harmonic and biharmonic maps. J. Korean Math. Soc. 54, 763–772 (2017)Google Scholar
  11. 11.
    Kamber, F.W., Tondeur, P.: Harmonic foliations. In: Proceedings of the National Science Foundation Conference on Harmonic Maps, Tulance, Dec 1980, Lecture Notes in Mathematics, 949, pp. 87–121. Springer, New York (1982)Google Scholar
  12. 12.
    Kamber, F.W., Tondeur, P.: Infinitesimal automorphisms and second variation of the energy for harmonic foliations. Tôhoku Math. J. 34, 525–538 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kamber, F.W., Tondeur, P.: De Rham-Hodge theory for Riemannian foliations. Math. Ann. 277, 415–431 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kashiwada, T.: On conformal Killing tensor. Nat. Sci. Rep. Ochanomizu Univ. 19, 67–74 (1968)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kitahara, H.: Remarks on square-integrable basic cohomology spaces on a foliated Riemannian manifold. Kodai Math. J. 2, 187–193 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Moroianu, A., Semmelmann, U.: Twistor forms on Kähler manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. II, 823–845 (2003)zbMATHGoogle Scholar
  17. 17.
    Nishikawa, S., Tondeur, P.: Transversal infinitesimal automorphisms for harmonic Kähler foliations. Tôhoku Math. J. 40, 599–611 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nishikawa, S., Tondeur, P.: Transversal infinitesimal automorphisms of harmonic foliations on complete manifolds. Ann. Glob. Anal. Geom. 7, 47–57 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Park, E., Richardson, K.: The basic Laplacian of a Riemannian foliation. Am. J. Math. 118, 1249–1275 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Semmelmann, U.: Conformal Killing forms on Riemannian manifolds. Math. Z. 245, 503–527 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tachibana, S.: On conformal Killing tensor in a Riemannian space. Tôhoku Math. J. 21, 56–64 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tondeur, P.: Geometry of Foliations. Birkhäuser, Basel (1997)CrossRefzbMATHGoogle Scholar
  23. 23.
    Yorozu, S.: Conformal and Killing vector fields on complete non-compact Riemannian manifolds. Adv. Stud. Pure Math. 3, 459–472 (1984). Geometry of Geodesics and Related TopicsGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsJeju National UniversityJejuKorea
  2. 2.Department of MathematicsNortheastern UniversityShenyangPeople’s Republic of China

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