A metric characterization of Riemannian submersions
According to the first author [Sib. Math. J. 28, No. 4, 552-562 (1987; Zbl 0643.53053); translation from Sib. Mat. Zh. 28, No. 4(164), 44-56 (1987)], a map between metric spaces is a submetry if it sends closed balls around a point to closed balls of the same radius, around the image point. In the paper under review, one proves that a submetry between smooth Riemannian manifolds is a \(C^{1,1}\) Riemannian submersion. The \(C^{1,1}\) regularity is optimal, since there is a submetry on the hyperbolic plane \(\mathbb{H}^{2}\rightarrow \mathbb{R}\) that is not \( C^{2}\). In the case of the metric projection of an open Riemannian manifold with nonnegative sectional curvature onto its soul, it is known that the projection is \(C^{2}\) and a.e. smooth [\textit{G. Perelman}, J. Differ. Geom. 40, 209-212 (1994; Zbl 0818.53056)]. The authors prove that if \(\varphi :M\rightarrow N\) is a submetry of complete Riemannian manifolds and \(M\) has a nonnegative sectional curvature, then \(N\) has also a nonnegative sectional curvature and if \(N\) is compact and \(M\) is flat, then \(N\) is flat. Another result of the paper uses the Cheeger-Gromoll splitting theorem [\textit{J. Cheeger} and \textit{D. Gromoll}, J. Differ. Geom. 6, 119-128 (1971; Zbl 0223.53033)] and states that if \(\varphi :M\rightarrow N\) is a submetry of complete Riemannian manifolds which have nonnegative Ricci curvatures, then there is a submetry \(\varphi _{0}:M_{0}\rightarrow N_{0}\) such that \(M\) and \(N\) are isometric with \(M_0\times \mathbb{R}^{n_{0}}\) and \(N_0\times \mathbb{R}^{n_{0}}\) respectively and \(\varphi \) can be identified with \( (\varphi _{0}\times Id)\). Defining the submetry with a soul as a submetry \(\varphi :M\rightarrow N\) for which there is a non-distance increasing map \(\psi :N\rightarrow M\) with \( \varphi \circ \psi =Id_{N}\), the authors prove some interesting results in the case where \(N\) is compact, extending obviously the case of the metric projection of an open Riemannian manifold with nonnegative sectional curvature onto its soul. For example, there is proved that: \(\psi (N)\) is a totally geodesic submanifold of \(M\); if \(M\) is a complete manifold with nonnegative sectional curvature, then \(\varphi \) is a \(C^{2}\) Riemannian submersion; if \(M\) has a positive sectional curvature then \(N\) is a point or \(N=M\) and \(\varphi \) is an isometry. A new proof of a result involving the warp product of two Riemannian manifolds, which appears in [\textit{G. Walschap}, J. Geom. Anal. 2, No. 4, 373-381 (1992; Zbl 0769.53021)] is also given.
scientific article; zbMATH DE number 1563295