2025/05/11 by López, Jesús A. Álvarez · 2 citations
#57R30 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.06957
For a Riemannian foliation F on a compact manifold M with a bundle-like metric, the de Rham complex of M is C∞-splitted as the direct sum of the basic complex and its orthogonal complement. Then the basic component κb 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.