Twisted submersions in nonnegative sectional curvature
Let \(F\) be a metric foliation with complete leaves on a Riemannian manifold \((M,g)\) of nonnegative sectional curvature. A curve is said to be horizontal if it is orthogonal to the leaves of \(F\) at every point. Given any point \(P\in M\), consider the subset of \(M\) that can be reached by the horizontal curves from \(P\); the work of \textit{B. Wilking} [Geom. Funct. Anal. 17, No. 4, 1297--1320 (2007; Zbl 1139.53014)] shows that they form a singular metric foliation \(F^{\#}\) which is called the dual foliation; if the metric is positively curved, then the dual foliation contains only a single leaf. The authors consider the case in which the foliation appears as the collection of fibers of a Riemannian submersion; a Riemannian submersion \(\pi:M\rightarrow B\) with fiber \(F\) is said to be twisted if the dual foliation only contains one leaf. The authors show: \smallbreak\noindent{Theorem 1.} Let \(\pi:M\rightarrow B\) be a Riemannian submersion. Assume that \(M\) is complete and has nonnegative sectional curvature. Let \(\delta\) be the connecting homomorphism from \(\pi_q(B)\) to \(\pi_{q-1}(F)\) in the long exact sequence in homotopy. If \(\pi\) is not twisted then any element in the range of \(\delta\) can be represented by a map \(f\) from \(S^{q-1}\) to the fiber \(F\) which is not surjective. \smallbreak\noindent Various applications of this result are given, for example: \smallbreak\noindent{Theorem 3.} Let \(M\) be a compact simply connected manifold. Assume that \(N=M\times S^2\) admits a nonnegatively curved metric such that there exists a Riemannian submersion \(\pi:N\rightarrow M\) with totally geodesic fibers. Then either \(\pi\) is twisted or \(N\) is isometric to the Riemannian product of \(M\times S^2\). \noindent The homotopy conditions in Theorem 1 can be reformulated in cohomological terms using transgressive elements.
scientific article; zbMATH DE number 6210419
scientific article