The basic component of the mean curvature of Riemannian foliations

TL;DR

This paper investigates the mean curvature of Riemannian foliations, establishing invariants, conditions for minimal leaves, and implications for the topology of the manifold, with new insights into the basic cohomology class and tautness.

Contribution

It introduces the basic cohomology class of the mean curvature form as an invariant and characterizes when leaves are minimal, extending understanding of Riemannian foliations.

Findings

The basic component of the mean curvature form is closed and defines an invariant class.

The class vanishes iff there exists a bundle-like metric with minimal leaves.

Foliations with positive transverse Ricci curvature or codimension one are taut.

Abstract

For a Riemannian foliation $F$ on a compact manifold $M$ with a bundle-like metric, the de Rham complex of $M$ is $C^{\infty}$-splitted as the direct sum of the basic complex and its orthogonal complement. Then the basic component $\kappa_{b}$ of the mean curvature form of $F$ is closed and defines a class $\xi(F)$ in the basic cohomology that is invariant under any change of the bundle-like metric. Moreover, any element in $\xi(F)$ can be realized as the basic component of the mean curvature of some bundle-like metric. It is also proved that $\xi(F)$ vanishes iff there exists some bundle-like metric on $M$ for which the leaves are minimal submanifolds. As a consequence, this tautness property is verified in any of the following cases: (a) when the Ricci curvature of the transverse Riemannian structure is positive, or (b) when $F$ is of codimension one. In particular, a compact manifold with a Riemannian foliation of codimension one has infinite fundamental group. A small correction of a lemma from the original manuscript is included as an addendum, written in collaboration with Ken Richardson.