Generalized Obata theorem and its applications on foliations

TL;DR

This paper extends the Obata theorem to foliated manifolds, characterizing when such manifolds are transversally isometric to a sphere based on the existence of specific basic functions, and classifies manifolds with transversal conformal fields.

Contribution

The authors prove a generalized Obata theorem for foliations, providing a new characterization of foliated manifolds that are transversally spherical and classifying those with transversal conformal fields.

Findings

Foliated manifolds satisfying the basic function condition are transversally isometric to a sphere.

The theorem characterizes the geometric structure of foliations via basic functions and their differential properties.

Classification of manifolds with transversal non-isometric conformal fields is achieved.

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.