A NOTE ON GENERALIZED LICHNEROWICZ-OBATA THEOREMS FOR RIEMANNIAN FOLIATIONS

Title & Authors
A NOTE ON GENERALIZED LICHNEROWICZ-OBATA THEOREMS FOR RIEMANNIAN FOLIATIONS
Pak, Hong-Kyung; Park, Jeong-Hyeong;

Abstract
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 $\small{{\lambda}}$-automorphism with ${\lambda} Keywords Riemannian foliation;generalized Lichnerowicz-Obata theorem;$\small{{\lambda}}$-automorphism;transversally Einstein foliation; Language English Cited by 1. A NOTE ON THE GENERALIZED OBATA THEOREM FOR RIEMANNIAN FOLIATIONS,; Advanced Studies in Contemporary Mathematics, 2012. vol.22. 3, pp.459-465 1. The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds, Nonlinear Analysis, 2015, 126, 262 2. A Lower Bound on the Spectrum of the Sublaplacian, The Journal of Geometric Analysis, 2015, 25, 3, 1492 References 1. D. Dominguez, Finiteness and tenseness theorems for Riemannian foliations, Amer. J. Math. 120 (1998), no. 6, 1237-1276. 2. G. Habib, Eigenvalues of the basic Dirac operator on quaternion-Kahler foliations, Ann. Global Anal. Geom. 30 (2006), no. 3, 286-298. 3. M. J. Jung and S. D. Jung, Riemannian foliations admitting transversal conformal fields, Geom. Dedicata 133 (2008), 155-168. 4. S. D. Jung, K. R. Lee, and K. Richardson, Generalized Obata theorem and its applications on foliations, ArXiv:0908.4545v1 [math.DG], 2009. 5. J. Lee and K. Richardson, Lichnerowicz and Obata theorems for foliations, Pacific J. Math. 206 (2002), no. 2, 339-357. 6. A. Lichnerowicz, Geometrie des groupes de transformations, Travaux et Recherches Mathematiques III, Dunod, Paris, 1958. 7. M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333-340. 8. H. K. Pak,$\lambda$-automorphisms of a Riemannian foliation, Ann. Global Anal. Geom. 13 (1995), no. 3, 281-288. 9. H. K. Pak,$\lambda\$-automorphisms of a Riemannian foliation II, Geom. Dedicata 66 (1997), no. 1, 19-25.

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H. K. Pak, Geometry of conformal transformations of foliated Riemannian manifolds, Adv. Stud. Contemp. Math. 9 (2004), no. 1, 33-40.

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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.

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