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http://dx.doi.org/10.4134/JKMS.2012.49.5.977

ON TRANSVERSALLY HARMONIC MAPS OF FOLIATED RIEMANNIAN MANIFOLDS  

Jung, Min-Joo (Department of Mathematics Jeju National University)
Jung, Seoung-Dal (Department of Mathematics and Research Institute for Basic Sciences Jeju National University)
Publication Information
Journal of the Korean Mathematical Society / v.49, no.5, 2012 , pp. 977-991 More about this Journal
Abstract
Let (M,F) and (M',F') be two foliated Riemannian manifolds with M compact. If the transversal Ricci curvature of F is nonnegative and the transversal sectional curvature of F' is nonpositive, then any transversally harmonic map ${\phi}:(M,F){\rightarrow}(M^{\prime},F^{\prime})$ is transversally totally geodesic. In addition, if the transversal Ricci curvature is positive at some point, then ${\phi}$ is transversally constant.
Keywords
transversal tension field; transversally harmonic map; normal variational formula; generalized Weitzenb$\ddot{o}$ck type formula;
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1 B. O'Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459-469.   DOI
2 H. K. Pak and J. H. Park, Transversal harmonic transformations for Riemannian foliations, Ann. Global Anal. Geom. 30 (2006), no. 1, 97-105.   DOI
3 E. Park and K. Richardson, The basic Laplacian of a Riemannian foliation, Amer. J. Math. 118 (1996), no. 6, 1249-1275.   DOI
4 H. C. J. Sealey, Harmonic maps of small energy, Bull. London Math. Soc. 13 (1981), no. 5, 405-408.   DOI
5 Ph. Tondeur, Foliations on Riemannian Manifolds, New-York, Springer-Verlag, 1988.
6 Ph. Tondeur, Geometry of Foliations, Basel: Birkhauser Verlag, 1997.
7 Y. L. Xin, Geometry of Harmonic Maps, Birkhauser, Boston, 1996.
8 S. Yorozu and T. Tanemura, Green's theorem on a foliated Riemannian manifold and its applications, Acta Math. Hungar. 56 (1990), no. 3-4, 239-245.   DOI
9 J. A. Alvarez Lopez, The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom. 10 (1992), no. 2, 179-194.   DOI
10 J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 106-160.
11 S. D. Jung, The first eigenvalue of the transversal Dirac operator, J. Geom. Phys. 39 (2001), no. 3, 253-264.   DOI   ScienceOn
12 S. D. Jung, K. R. Lee, and K. Richardson, Generalized Obata theorem and its applications on foliations, J. Math. Anal. Appl. 376 (2011), no. 1, 129-135.   DOI   ScienceOn
13 F. W. Kamber and Ph. Tondeur, Infinitesimal automorphisms and second variation of the energy for harmonic foliations, Tohoku Math. J. (2) 34 (1982), no. 4 525-538.   DOI
14 P. Molino, Riemannian Foliations, translated from the French by Grant Cairns, Boston: Birkhaser, 1988.
15 J. Konderak and R. Wolak, Transversally harmonic maps between manifolds with Riemannian foliations, Q. J. Math. 54 (2003), no. 3, 335-354.   DOI
16 J. Konderak and R. Wolak, Some remarks on transversally harmonic maps, Glasg. Math. J. 50 (2008), no. 1, 1-16.