Riemannian foliations admitting transversal conformal fields II

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• 作者：Seoung Dal Jung (1)
  
  1. Department of Mathematics ; Jeju National University ; Jeju ; 690-756 ; Korea
  
• 关键词：Riemannian foliation ; Transversal conformal field ; Generalized Obata theorem ; 53C20 ; 57R30
• 刊名：Geometriae Dedicata
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：175
• 期：1
• 页码：257-266
• 全文大小：163 KB
• 参考文献：1. Alvarez L贸pez, JA (1992) The basic component of the mean curvature of Riemannian foliations. Ann. Glob. Anal. Geom. 10: pp. 179-194 CrossRef
  2. Jung, SD (2001) The first eigenvalue of the transversal Dirac operator. J. Geom. Phys. 39: pp. 253-264 CrossRef
  3. Jung, SD (2007) Eigenvalue estimates for the basic Dirac operator on a Riemannian foliation admitting a basic harmonic 1-form. J. Geom. Phys. 57: pp. 1239-1246 CrossRef
  4. Jung, MJ, Jung, SD (2008) Riemannian foliations admitting transversal conformal fields. Geom. Dedic. 133: pp. 155-168 CrossRef
  5. Jung, S.D., Lee, K.R., Richardson, K.: Generalized Obata theorem and its applications on foliations. J. Math. Anal. Appl. 376, 129鈥?35 (2011)
  6. Kamber, F.W., Tondeur, Ph.: Harmonic foliations. Proceedings of the National Science Foundation Conference on Harmonic Maps, Tulane, December 1980. Lecture Notes in Mathematics, vol. 949, pp. 87鈥?21. Springer, New-York (1982)
  7. Kamber, F.W., Tondeur, Ph.: Infinitesimal automorphisms and second variation of the energy for harmonic foliations. Tohoku Math. J. 34, 525鈥?38 (1982)
  8. Lee, J, Richardson, K (2002) Lichnerowicz and Obata theorems for foliations. Pac. J. Math. 206: pp. 339-357 CrossRef
  9. March, P, Min-Oo, M, Ruh, EA (1996) Mean curvature of Riemannian foliations. Can. Math. Bull. 39: pp. 95-105 CrossRef
  10. Mason, P (2000) An application of stochastic flows to Riemannian foliations. Houst. J. Math. 26: pp. 481-515
  11. Pak, JS, Yorozu, S (1988) Transverse fields on foliated Riemannian manifolds. J. Korean Math. Soc. 25: pp. 83-92
  12. Park, JH, Yorozu, S (1993) Transversal conformal fields of foliations. Nihonkai Math. J. 4: pp. 73-85
  13. Tondeur, P, Toth, G (1987) On transversal infinitesimal automorphisms for harmonic foliations. Geom. Dedic. 24: pp. 229-236 CrossRef
  14. Yano, K (1966) On Riemannian manifolds with constant scalar curvature admitting a conformal transformation group. Proc. Nat. Acad. Sci. USA 55: pp. 472-476 CrossRef
  15. Yorozu, S, Tanemura, T (1990) Green鈥檚 theorem on a foliated Riemannian manifold and its applications. Acta Math. Hung. 56: pp. 239-245 CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1572-9168

文摘

Let \((M,g_M,{\mathcal {F}})\) be a closed, oriented Riemannian manifold with a foliation \({\mathcal {F}}\) of codimension \(q\) and a bundle-like metric \(g_M\) . Assume that the transversal scalar curvature is non-zero constant. If \(M\) admits a transversal conformal field satisfying some conditions, then \({\mathcal {F}}\) is transversally isometric to a sphere.