A note on transverse divergence and taut Riemannian foliations

TL;DR

This paper characterizes taut Riemannian foliations via transverse divergence, providing new tools for standard examples and deriving classical tautness results, with additional insights into transversally oriented foliations and submanifold orthogonality.

Contribution

It introduces a new characterization of taut Riemannian foliations using transverse divergence and applies it to various specific cases.

Findings

Characterization of taut Riemannian foliations via transverse divergence

Application to standard examples of foliations

Derivation of Haefliger's tautness result in this setting

Abstract

In this note we give a characterization of taut Riemannian foliations using the transverse divergence. This result turns out to be a convenient tool in the case of some standard examples. Furthermore, we show that a classical tautness result of Haefliger can be obtained in our particular setting as a straightforward consequence. In the final part of the paper we obtain a tautness characterization for transversally oriented foliations with dense leaves and investigate the case of a submanifold of lower dimension which is closed and orthogonal to the leaves.