The basic component of the mean curvature of Riemannian foliations

✍ Scribed by Jesús A. Alvarez López

• Publisher: Springer
• Year: 1992
• Tongue: English
• Weight: 746 KB
• Volume: 10
• Category: Article
• ISSN: 0232-704X
• DOI: 10.1007/bf00130919

✦ Synopsis

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 cb of the mean curvature form of F is closed and defines a class (Y) in the basic cohomology that is invariant under any change of the bundle-like metric. Moreover, any element in C(F) can be realized as the basic component of the mean curvature of some bundle-like metric.

It is also proved that C(TF) 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 YF is of codimension one. In particular, a compact manifold with a Riemannian foliation of codimension one has infinite fundamental group.