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arXiv · 0908.4545

Generalized Obata theorem and its applications on foliations

Seoung Dal Jung·Keum Ran Lee·Ken Richardson
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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 $q>1$ and a bundle-like metric. Then $(M, F)$ is transversally isometric to the q-sphere of radius 1/c in (q+1)-dimensional Euclidean space endowed with the action of a discrete subgroup of the orthogonal group O(q), if and only if there exists a non-constant basic function f such that $\nabla_X df = -c^2 f X^\flat for all basic normal vector fields X, where c is a positive constant and \nabla 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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Seoung Dal Jung, Keum Ran Lee, Ken Richardson. 2009-08-31. Generalized Obata theorem and its applications on foliations. https://doi.org/10.1016/j.jmaa.2010.10.022

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