A metric characterization of Riemannian submersions for A. D. Alexandrov manifolds of bounded curvature
The author proves that each submetry from one manifold with (bilaterally) bounded curvature on the sense of A. D. Alexandrov into another is a Riemannian \(C^{1,1}\)-submersion with respect to the distance coordinates. Conversely, each Riemannian \(C^1\)-submersion of the complete A. D. Alexandrov manifolds with bounded curvature is a submetry. These assertions characterize Riemannian submersions for manifolds with bounded curvature and generalize the results obtained by \textit{V. N. Berestovskij} and \textit{L. Guijarro} [Ann. Global Anal. Geom. 18, No. 6, 577--588 (2000; Zbl 0992.53025)] for smooth Riemannian manifolds. Some questions related to these problems are also discussed. This article is an English translation of the author's contribution in the book [\textit{Yu. G. Reshetnyak} (ed.) et al., Proceedings of the conference `Geometry and applications' dedicated to the 70th anniversary of Prof. Victor Toponogov. Novosibirsk: Izdatel'stvo Instituta Matematiki (2001; Zbl 0987.53016)].
scientific article; zbMATH DE number 2097420