Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alvarez López, J.A.: The basic component of the mean curvature of Riemannian foliations. Ann. Glob. Anal. Geom. 10, 179–194 (1992)
Aoki, T., Yorozu, S.: \(L^2\)-transverse conformal and Killing fields on complete foliated Riemannian manifolds. Yokohama Math. J. 36, 27–41 (1988)
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)
Jung, S.D.: Transverse Killing forms on complete foliated Riemannian manifolds. Honam Math. J. 36, 731–737 (2014)
Jung, S.D.: Transverse conformal Killing forms on Kähler foliations. J. Geom. Phys. 90, 29–41 (2015)
Jung, M.J., Jung, S.D.: Riemannian foliations admitting transversal conformal fields. Geom. Dedicata 133, 155–168 (2008)
Jung, S.D., Jung, M.J.: Transverse Killing forms on a Kähler foliation. Bull. Korean Math. Soc. 49, 445–454 (2012)
Jung, S.D., Richardson, K.: Transverse conformal Killing forms and a Gallot–Meyer theorem for foliations. Math. Z. 270, 337–350 (2012)
Jung, S.D., Jung, M.J.: Transversally holomorphic maps between Kähler foliations. J. Math. Anal. Appl. 416, 683–697 (2014)
Jung, M.J., Jung, S.D.: Liouville type theorems for transversally harmonic and biharmonic maps. J. Korean Math. Soc. 54, 763–772 (2017)
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)
Kamber, F.W., Tondeur, P.: Infinitesimal automorphisms and second variation of the energy for harmonic foliations. Tôhoku Math. J. 34, 525–538 (1982)
Kamber, F.W., Tondeur, P.: De Rham-Hodge theory for Riemannian foliations. Math. Ann. 277, 415–431 (1987)
Kashiwada, T.: On conformal Killing tensor. Nat. Sci. Rep. Ochanomizu Univ. 19, 67–74 (1968)
Kitahara, H.: Remarks on square-integrable basic cohomology spaces on a foliated Riemannian manifold. Kodai Math. J. 2, 187–193 (1979)
Moroianu, A., Semmelmann, U.: Twistor forms on Kähler manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. II, 823–845 (2003)
Nishikawa, S., Tondeur, P.: Transversal infinitesimal automorphisms for harmonic Kähler foliations. Tôhoku Math. J. 40, 599–611 (1988)
Nishikawa, S., Tondeur, P.: Transversal infinitesimal automorphisms of harmonic foliations on complete manifolds. Ann. Glob. Anal. Geom. 7, 47–57 (1989)
Park, E., Richardson, K.: The basic Laplacian of a Riemannian foliation. Am. J. Math. 118, 1249–1275 (1996)
Semmelmann, U.: Conformal Killing forms on Riemannian manifolds. Math. Z. 245, 503–527 (2003)
Tachibana, S.: On conformal Killing tensor in a Riemannian space. Tôhoku Math. J. 21, 56–64 (1969)
Tondeur, P.: Geometry of Foliations. Birkhäuser, Basel (1997)
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 Topics
Acknowledgements
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).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jung, S.D., Liu, H. \(L^\mathrm{2}\)-transverse conformal Killing forms on complete foliated manifolds. Math. Z. 288, 665–677 (2018). https://doi.org/10.1007/s00209-017-1905-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-017-1905-0