The basic component of the mean curvature of Riemannian foliations
Jesús A. Álvarez López
TL;DR
The paper defines the basic component $\kappa_b$ of the mean curvature for Riemannian foliations on compact manifolds and proves it is a closed basic form whose class $\xi(\mathcal{F})=[\kappa_b]$ is invariant under metric deformations. It shows a precise orthogonal splitting of the de Rham complex into basic and non-basic parts, and derives a basic Hodge theory that connects tautness with the vanishing of $\xi(\mathcal{F})$ and with nontrivial basic cohomology $H_b^q(\mathcal{F})$. The results yield tautness under positive transverse Ricci curvature or in codimension-one, and imply strong topological consequences like infinite fundamental group for codimension-one foliations. A reciprocal construction demonstrates the realizability of all elements of $\xi(\mathcal{F})$ and the invariance of $[\kappa_b]$ under broad metric changes.
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 $κ_{b}$ of the mean curvature form of $F$ is closed and defines a class $ξ(F)$ in the basic cohomology that is invariant under any change of the bundle-like metric. Moreover, any element in $ξ(F)$ can be realized as the basic component of the mean curvature of some bundle-like metric. It is also proved that $ξ(F)$ 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.