Bounded Riemannian submersions

In this paper, we establish global metric properties of a Riemannian submersion pi : Mn+k --> B-n for which the undamental tensors are bounded in norm: /A/ less than or equal to C-A, /T/ less than or equal to C-T. For example, if B is compact and simply connected, then there exists a constant C = C(B, C-A, C-T, k) such that for all p is an element of B, d(Fp) less than or equal to C.d(M), where d(Fp) denotes the intrinsic distance function on the fiber F-p := pi (-1)(p), and d(M) denotes the distance function of M restricted to F-p. When applied to the metric projection pi : M --> Sigma from an open manifold of nonnegative curvature M onto its soul C, this property implies that the ideal boundary of M can be determined from a single fiber of the projection. As a second application, we show that there are only finitely many isomorphism types of fiber bundles among the class of Riemannian submersions whose base space and total space both satisfy fixed geometric bounds (volume from below, diameter from above, curvature from above and below).