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Finiteness theorems for submersions and souls

Author: Kristopher Tapp
Journal: Proc. Amer. Math. Soc. 130 (2002), 1809-1817
MSC (1991): Primary 53C20
Published electronically: October 12, 2001
MathSciNet review: 1887030
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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.

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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

Keywords: Soul, vector bundle, finiteness theorem, Riemannian submersion
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
Article copyright: © Copyright 2001 American Mathematical Society

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