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Generalized Obata theorem and its applications on foliations
Authors:Seoung Dal Jung  Keum Ran Lee
Institution: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
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 View the MathML source 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.
Keywords:The generalized Obata theorem  Transversal Killing field  Transversal conformal field
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