# Generalized Obata theorem and its applications on foliations

Seoung Dal Jung, Keum Ran Lee, Ken Richardson

Arxiv ID: 0908.4545•Last updated: 1/28/2021

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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