Generalized Obata theorem and its applications on foliations

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 g M . Then ( M , F ) is transversally isometric to ( S q ( 1 / c ) , G ) , where S q ( 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 ∇ X d f = − c 2 f X b 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.