Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-19T12:08:03.575Z Has data issue: false hasContentIssue false
  • English
  • Français

TRANSVERSAL INFINITESIMAL AUTOMORPHISMS ON KÄHLER FOLIATIONS

Part of: Global differential geometry

Published online by Cambridge University Press:  02 February 2012

SEOUNG DAL JUNG  and
HUILI LIU
SEOUNG DAL JUNG*
Affiliation:
Department of Mathematics and Research Institute for Basic Sciences, Jeju National University, Jeju 690-756, Korea (email: sdjung@jejunu.ac.kr)
HUILI LIU
Affiliation:
Department of Mathematics, Northeastern University, 110004 Shenyang, PR China (email: liuhl@mail.neu.edu.cn)
*
✉For correspondence; e-mail: sdjung@jejunu.ac.kr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ℱ be a Kähler foliation on a compact Riemannian manifold M. If the transversal scalar curvature of ℱ is nonzero constant, then any transversal conformal field is a transversal Killing field; and if the transversal Ricci curvature is nonnegative and positive at some point, then there are no transversally holomorphic fields.

Keywords

transversal Killing fieldtransversal conformal fieldtransversal Jacobi fieldtransversally holomorphic field

MSC classification

Secondary: 53C12: Foliations (differential geometric aspects)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 3 , December 2012 , pp. 405 - 415
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

The first author was supported by the National Research Foundation of Korea (NRF) Grant NRF-2011-616-C00040, and the second author was supported by NSFC no. 11111140377.

References

[1]Alvarez López, J. A., ‘The basic component of the mean curvature of Riemannian foliations’, Ann. Global Anal. Geom. 10 (1992), 179–194.CrossRefGoogle Scholar
[2]Jung, S. D., ‘Transversal infinitesimal automorphisms for non-harmonic Kähler foliation’, Far East J. Math. Sci. (FJMS) Special Volume (2000), Part II (Geometry and Topology) 169–177.Google Scholar
[3]Jung, S. D., ‘Eigenvalue estimates for the basic Dirac operator on a Riemannian foliation admitting a basic harmonic 1-form’, J. Geom. Phys. 57 (2007), 1239–1246.CrossRefGoogle Scholar
[4]Jung, M. J. and Jung, S. D., ‘Riemannian foliations admitting transversal conformal fields’, Geom. Dedicata 133 (2008), 155–168.CrossRefGoogle Scholar
[5]Jung, S. D., Lee, K. R. and Richardson, K., ‘Generalized Obata theorem and its applications on foliations’, J. Math. Anal. Appl. 376 (2011), 129–135.CrossRefGoogle Scholar
[6]Jung, S. D. and Richardson, K., ‘Transverse conformal Killing forms and a Gallot–Meyer theorem for foliations’, Math. Z., doi:10.1007/s00209-010-0800-8.CrossRefGoogle Scholar
[7]Kamber, F. W. and Tondeur, Ph., ‘Harmonic foliations’, in: Proc. National Science Foundation Conference on Harmonic Maps, Tulance, Dec. 1980, Lecture Notes in Mathematics, 949 (Springer, New York, 1982), pp. 87–121.Google Scholar
[8]Kamber, F. W. and Tondeur, Ph., ‘Infinitesimal automorphisms and second variation of the energy for harmonic foliations’, Tohoku Math. J. (2) 34 (1982), 525–538.CrossRefGoogle Scholar
[9]Nishikawa, S. and Tondeur, Ph., ‘Transversal infinitesimal automorphisms for harmonic Kähler foliations’, Tohoku Math. J. (2) 40 (1988), 599–611.CrossRefGoogle Scholar
[10]Pak, J. S. and Yorozu, S., ‘Transverse fields on foliated Riemannian manifolds’, J. Korean Math. Soc. 25 (1988), 83–92.Google Scholar
[11]Tashiro, Y., ‘On conformal and projective transformations in Kählerian manifolds’, Tohoku Math. J. (2) 14 (1962), 317–320.CrossRefGoogle Scholar
[12]Tondeur, Ph., Foliations on Riemannian Manifolds (Springer, New York, 1988).CrossRefGoogle Scholar
[13]Yano, K., ‘On harmonic and Killing vector fields’, Ann. of Math. (2) 55 (1952), 38–45.CrossRefGoogle Scholar
[14]Yorozu, S. and Tanemura, T., ‘Green’s theorem on a foliated Riemannian manifold and its applications’, Acta Math. Hungar. 56 (1990), 239–245.CrossRefGoogle Scholar