Generalized Obata theorem and its applications on foliations |
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Authors: | Seoung Dal Jung Keum Ran Lee |
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Affiliation: | a Department of Mathematics and Research Institute for Basic Sciences, Jeju National University, Jeju 690-756, Republic of Korea b Department of Mathematics, Jeju National University, Jeju 690-756, Republic of Korea c Department of Mathematics, Texas Christian University, Fort Worth, TX 76129, USA |
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Abstract: | We prove the generalized Obata theorem on foliations. Let M be a complete Riemannian manifold with a foliation F of codimension q?2 and a bundle-like metric gM. Then (M,F) is transversally isometric to (Sq(1/c),G), where Sq(1/c) is the q-sphere of radius 1/c in (q+1)-dimensional Euclidean space and G is a discrete subgroup of the orthogonal group O(q), if and only if there exists a non-constant basic function f such that for all basic normal vector fields X, where c is a positive constant and ∇ is the connection on the normal bundle. By the generalized Obata theorem, we classify such manifolds which admit transversal non-isometric conformal fields. |
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Keywords: | The generalized Obata theorem Transversal Killing field Transversal conformal field |
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