The basic component of the mean curvature of Riemannian foliations

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.