Submetries of space forms of nonnegative curvature
The author introduces the following notion of a submetry: a map \(p: M\to N\) of metric spaces is called a submetry if for every point \(x\in M\) the image of each closed ball with the center at x under the map p is the closed ball of the same radius with the center at the point p(x). Every Riemannian submersion of a complete Riemannian manifold is a submetry. The following main results are established. Theorem 1. Any submetry \(p: M\to N\) of complete Euclidean spaces M and N (possibly infinite dimensional) can be represented as a composition \(p=i_ 1\circ p_ 1\), where \(p_ 1\) is an orthogonal projection of M on a closed Euclidean subspace \(M_ 1\subset M\) and \(i_ 1: M_ 1\to N\) is an isometry. Theorem 2. Any submetry \(p: M\to N\) of unit spheres of complete Euclidean spaces is an isometry. Furthermore a connection between special Riemannian submersions and the existence of a transversal foliation of a given bundle is established in the paper. The author proves that the Hopf bundles do not have transversal foliations.
unit spheres of complete Euclidean spaces
scientific article; zbMATH DE number 4048589