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**Taut submanifolds and foliations.**
*(English)*
Zbl 1302.53032

S. Carter and A. West [Proc. Lond. Math. Soc. (3) 25, 701–720 (1972; Zbl 0242.53029)] called a submanifold \(L\) of a Euclidean space \(V\) taut if all the distance functions \(d_q^2 :L\to\mathbb{R}\), \(q\in V\) being non-focal for \(L\), are perfect Morse functions for some field \(\mathbb{F}\). Using different approaches K. Grove and S. Halperin [Arch. Math. 56, No. 3, 288–299 (1991; Zbl 0726.53030)] and C.-L. Terng and G. Thorbergsson [Math. Sci. Res. Inst. Publ. 32, 181–228 (1997; Zbl 0906.53043)] called a submanifold \(L\) of a complete Riemannian manifold \(M\) taut if all the energy functionals \(E_q\) on the space of \(H^1\)-pathes from \(L\) to a fixed non-focal for \(L\) point \(q\in M\) are perfect Morse functions for some field \(\mathbb{F}\). Here, the author calls a Riemannian singular foliation \(\mathcal{F}\) taut when all its leaves are taut and proves (among other results) the following: (1) if \(L\) is taut for some field \(\mathbb{F}\), then \(L\) is taut for \(\mathbb {Z}_2\), (2) tautness of a closed singular Riemannian foliation \(\mathcal{F}\) is a metric property of the leaf space \(M/\mathcal{F}\), (3) such a foliation \(\mathcal{F}\) is taut with the leaf space \(M/\mathcal{F}\) being an orbifold iff \(M/\mathcal{F}\) is isometric to the quotient \(N/\Gamma\), \(N\) being a pointwise taut Riemannian manifold and \(\Gamma\) a discrete group of isometries of \(N\).

(Note that D. Sullivan in [Comment. Math. Helv. 54, 218–223 (1979; Zbl 0409.57025)] called a foliation geometrically taut when all its leaves have zero mean curvature with respect to some fixed Riemannian structure of the ambient manifold. This condition occurs to be equivalent to another one called the topological tautness. It seems that Sullivan’s conditions are independent from the foliation tautness considered in the reviewed article.)

(Note that D. Sullivan in [Comment. Math. Helv. 54, 218–223 (1979; Zbl 0409.57025)] called a foliation geometrically taut when all its leaves have zero mean curvature with respect to some fixed Riemannian structure of the ambient manifold. This condition occurs to be equivalent to another one called the topological tautness. It seems that Sullivan’s conditions are independent from the foliation tautness considered in the reviewed article.)

Reviewer: Paweł Walczak (Łódź)