Finiteness theorems for submersions and souls
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Abstract:
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 a 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.References
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Additional Information
- Kristopher Tapp
- Affiliation: Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651
- MR Author ID: 630309
- Email: ktapp@math.sunysb.edu
- Received by editor(s): November 16, 2000
- Received by editor(s) in revised form: December 15, 2000
- Published electronically: October 12, 2001
- Communicated by: Wolfgang Ziller
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1809-1817
- MSC (1991): Primary 53C20
- DOI: https://doi.org/10.1090/S0002-9939-01-06244-X
- MathSciNet review: 1887030