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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2505.06957 |
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| _version_ | 1866915547626602496 |
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| author | López, Jesús A. Álvarez |
| author_facet | López, Jesús A. Álvarez |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06957 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The basic component of the mean curvature of Riemannian foliations López, Jesús A. Álvarez Differential Geometry 57R30 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. |
| title | The basic component of the mean curvature of Riemannian foliations |
| topic | Differential Geometry 57R30 |
| url | https://arxiv.org/abs/2505.06957 |