대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제48권4호
  • /
  • Pages.769-777
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  • 2011
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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A NOTE ON GENERALIZED LICHNEROWICZ-OBATA THEOREMS FOR RIEMANNIAN FOLIATIONS

  • 투고 : 2009.12.10
  • 발행 : 2011.07.31
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초록

It was obtained in [5] generalized Lichnerowicz and Obata theorems for Riemannian foliations, which reduce to the results on Riemannian manifolds for the point foliations. Recently in [3], they studied a generalized Obata theorem for Riemannian foliations admitting transversal conformal fields. Each transversal conformal field is a ${\lambda}$-automorphism with ${\lambda}=1-{\frac{2}{q}}$ in the sense of [8]. In the present paper, we extend certain results established in [3] and study Riemannian foliations admitting ${\lambda}$-automorphisms with ${\lambda}{\geq}1-{\frac{2}{q}}$.

키워드

참고문헌

  1. D. Dominguez, Finiteness and tenseness theorems for Riemannian foliations, Amer. J. Math. 120 (1998), no. 6, 1237-1276. https://doi.org/10.1353/ajm.1998.0048
  2. G. Habib, Eigenvalues of the basic Dirac operator on quaternion-Kahler foliations, Ann. Global Anal. Geom. 30 (2006), no. 3, 286-298.
  3. M. J. Jung and S. D. Jung, Riemannian foliations admitting transversal conformal fields, Geom. Dedicata 133 (2008), 155-168. https://doi.org/10.1007/s10711-008-9240-6
  4. S. D. Jung, K. R. Lee, and K. Richardson, Generalized Obata theorem and its applications on foliations, ArXiv:0908.4545v1 [math.DG], 2009.
  5. J. Lee and K. Richardson, Lichnerowicz and Obata theorems for foliations, Pacific J. Math. 206 (2002), no. 2, 339-357. https://doi.org/10.2140/pjm.2002.206.339
  6. A. Lichnerowicz, Geometrie des groupes de transformations, Travaux et Recherches Mathematiques III, Dunod, Paris, 1958.
  7. M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333-340. https://doi.org/10.2969/jmsj/01430333
  8. H. K. Pak, $\lambda$-automorphisms of a Riemannian foliation, Ann. Global Anal. Geom. 13 (1995), no. 3, 281-288. https://doi.org/10.1007/BF00773660
  9. H. K. Pak, $\lambda$-automorphisms of a Riemannian foliation II, Geom. Dedicata 66 (1997), no. 1, 19-25. https://doi.org/10.1023/A:1004935527915
  10. H. K. Pak, Geometry of conformal transformations of foliated Riemannian manifolds, Adv. Stud. Contemp. Math. 9 (2004), no. 1, 33-40.
  11. J. H. Park and S. Yorozu, Transverse fields preserving the transverse Ricci field of a foliation, J. Korean Math. Soc. 27 (1990), no. 2, 167-175.
  12. Ph. Tondeur, Geometry of Foliations, Monographs in Math., Birkhauser, 1997.

피인용 문헌

  1. The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds vol.126, 2015, https://doi.org/10.1016/j.na.2015.06.024
  2. A Lower Bound on the Spectrum of the Sublaplacian vol.25, pp.3, 2015, https://doi.org/10.1007/s12220-014-9481-6