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Tenseness of Riemannian flows. (Flots riemanniens étirés.) (English. French summary) Zbl 1325.53036

The authors show that any transversely complete Riemannian foliation \(\mathcal{F}\) of dimension \(1\) (Riemannian flow) on any (possibly non-compact) manifold \(M\) is tense (i.e., \(M\) admits a Riemannian metric such that the mean curvature form of \(\mathcal{F}\) is basic). This is a generalization of a result of D. Domínguez [Am. J. Math. 120, No. 6, 1237–1276 (1998; Zbl 0964.53019)] for Riemannian foliations in compact manifolds. The proof is motivated by the works of P. Molino and V. Sergiescu [Manuscripta Math. 51, No. 1-3, 145–161 (1985; Zbl 0585.53026)], producing a simpler proof than the original one of Domínguez [loc. cit.]. This is achieved after the completion of two intermediate results which are interesting by themselves:

\((\ast)\) For any transversely complete Riemannian flow \((M,\mathcal{F})\), either it is an \(\mathbb{R}\)-bundle or the closure of every leaf is compact.

\((\ast\ast)\) In a transversely complete Riemannian flow with a strongly tense metric which is not an \(\mathbb{R}\)-bundle, the mean curvature form is given by the logarithm of the holonomy homomorphism \(\pi_1(M)\to\mathbb{R}\) of the determinant line bundle of Molino’s commuting shift. Showing that the class of the mean curvature form is independent on the chosen metric.

The paper is subdivided in five sections. As usual, the first section is devoted to present the main results and the basic background needed for a good comprehension of the work. It is completed with the description of direct applications of the main theorems which serves as an additional motivation. The authors state the twisted Poincaré duality of the basic cohomology for Riemannian flows on non-compact manifolds, this can be seen as an extension of the original result for compact manifolds given by F. W. Kamber and P. Tondeur [in: Structure transverse des feuilletages, Toulouse 1982, Astérisque 116, 108–116 (1984; Zbl 0559.58022)]. As a corollary, it is obtained a characterization of tautness of Riemannian flows on non-compact manifolds in terms of the basic cohomology in the spirit of X. Masa’s result [Comment. Math. Helv. 67, No. 1, 17–27 (1992; Zbl 0778.53029)]. The authors also observe that the Euler class and the Gysin sequence of Riemannian flows are obtained using the tenseness in the compact case, so both Euler class and Gysin sequence can be defined as a direct corollary of the main theorem.

The necessary machinery is introduced in the second section: tautness, tenseness, characteristic forms and Rummler’s formula for the mean curvature form.

The third section is completely devoted to prove Theorem \((\ast)\). This is a key point since it allows to subdivide the problem between \(\mathbb{R}\)-bundles, where all become easy, and Riemannian flows where the closure of every leaf is compact and so the classical techniques on compact manifolds will work.

The fourth section is the core of the paper. Under the hypothesis of compact closure of leaves the authors show that the problem can be reduced to linearly foliated torus bundles (which is a version of Molino’s theory to Riemannian flows). This is given at the level of the orthogonal frame bundle over the normal bundle of \(\mathbb{F}\), producing an \(O(\mathrm{codim}\mathcal{F})\)-invariant flow. The authors show that an \(O(\mathrm{codim}\mathcal{F})\)-invariant tenseness of this lifted flow implies the tenseness of the original Riemannian flow. The next step reduces the structure group to the semidirect product of linear maps with toral maps preserving the linear foliation. This allows to work with a much simpler structure group without loss of generality. Lemma 4.7 describes tenseness and tautness in this framework. Finally, Theorem \((\ast\ast)\) is proved.

The fifth section shows that every linearly foliated torus bundle with a leaf preserving action of a compact Lie group \(G\) admits a \(G\)-invariant strongly tense metric. Although there are many technical details in the proof, the key point for this result is given by the fact that we can obtain a mean of a characteristic form via integration relative to the Haar measure of \(G\), this produces a new \(G\)-invariant characteristic form in the spirit of the original work of J. Álvarez López [Ann. Global Anal. Geom. 10, No. 2, 179–194 (1992; Zbl 0759.57017)]. This last result, combined with the work in the previous section, completes the proof of the main theorem.
The paper is well written and directed to those researchers focused in Riemannian foliations and basic cohomology on non-compact manifolds.

MSC:

53C12 Foliations (differential geometric aspects)
57R30 Foliations in differential topology; geometric theory
37C10 Dynamics induced by flows and semiflows
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References:

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