Finiteness Theorems for Submersions and Souls

The first section of this paper provides an improvement upon known finiteness theorems for Riemannian submersions; that is, theorems which conclude that there are only finitely many isomorphism types of fiber bundles among Riemannian submersions whose total spaces and base spaces both satisfy certain geometric bounds. The second section of this paper provides a sharpening of some recent theorems which conclude that, for an open manifold of nonnegative curvature satisfying certain geometric bounds near its soul, there are only finitely many possibilities for the isomorphism class of normal bundle of the soul. A common theme to both sections is a reliance on basic facts about Riemannian submersions whose A and T tensors are both bounded in norm. 1. Counting Riemannian Submersions In this section we improve upon known theorems which say that there are only finitely many equivalence classes of Riemannian submersions whose base space and total space both satisfy fixed geometric bounds. We consider two Riemannian submersions, π1 : M1 → B1 and π2 : M2 → B2, to be C-equivalent if there exists a C diffeomorphism f : M1 → M2 which maps the fibers of π1 C-diffeomorphically onto the fibers of π2. Every Riemannian submersion is a fiber bundle (as long as the total space is complete), and C0-equivalence just means equivalence up to fiber bundle isomorphism. The following theorem was proven by the author: Theorem 1 ([10, Theorem 5.1]). Let n, k ∈ Z and V,D, λ ∈ R. Then there are only finitely many C1-equivalence classes in the set of Riemannian submersions π : Mn+k → B for which: (1) vol(B) ≥ V, diam(B) ≤ D, |sec(B)| ≤ λ. (2) vol(M) ≥ V, diam(M) ≤ D, |sec(M)| ≤ λ. (3) B is simply connected Here “vol”, “diam”, and “ sec” are shorthand for volume, diameter, and sectional curvature. We improve this theorem by dropping condition 3 and also (at the cost of proving only C0-finiteness) dropping the upper curvature bound on M : Theorem 2. Let n, k ∈ Z and V,D, λ ∈ R. Assume k ≥ 4. Then there are only finitely many C0-equivalence classes in the set of Riemannian submersions π : Mn+k → B for which: (1) vol(B) ≥ V, diam(B) ≤ D, |sec(B)| ≤ λ. (2) vol(M) ≥ V, diam(M) ≤ D, sec(M) ≥ −λ. Date : September 2, 2002. 1991 Mathematics Subject Classification. 53C20.