Tenseness of Riemannian flows

TL;DR

This paper proves that transversally complete Riemannian one-dimensional foliations on possibly non-compact manifolds are tense, extending Dominguez's result from compact to non-compact cases, with a simpler proof and applications to tautness characterization.

Contribution

It generalizes the tense property of Riemannian foliations to non-compact manifolds and simplifies the proof using Molino and Sergiescu's results.

Findings

Transversally complete Riemannian flows are tense.

Extension of Dominguez's result to non-compact manifolds.

Simplified proof approach.

Abstract

We show that any transversally complete Riemannian foliation F of dimension one on any possibly non-compact manifold M is tense; namely, (M,F) admits a Riemannian metric such that the mean curvature form of F is basic. This is a partial generalization of a result of Dominguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Dominguez. As an application, we generalize some well known results including Masa's characterization of tautness.