Bounded Riemannian submersions

Tapp, Kristopher

doi:10.1512/iumj.2000.49.1705

Cited by 12 publications

(25 citation statements)

References 9 publications

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“…Corollary 1.2 was obtained in [9], as a consequence of Theorem 1.1, which generalizes the fiber bundle finiteness results in [16,17] (cf. [18][19][20]).…”

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confidence: 71%

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Stability of almost submetries

Rong

2010

Front. Math. China

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In this paper, we consider a triple of Gromov-Hausdorff convergence:where A i are compact Alexandrov n-spaces and B i are compact Riemannian m-manifolds such that the curvature, diameter and volume are suitably bounded (non-collapsing). When f is a submetry, we give a necessary and sufficient condition for the sequence to be stable, that is, for i large, there are homeomorphisms, Ψ i :When f is an ε-submetry with ε > 0, we obtain a sufficient condition for the stability in the case that A i are Riemannian manifolds. Our results generalize the stability/finiteness results on fiber bundles by Riemannian submersions and by submetries.

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“…Corollary 1.2 was obtained in [9], as a consequence of Theorem 1.1, which generalizes the fiber bundle finiteness results in [16,17] (cf. [18][19][20]).…”

supporting

confidence: 71%

“…Theorems C and D generalize the stability/finiteness results in [16] (cf. [18,19]) where f i is a Riemannian submersion.…”

supporting

confidence: 58%

Stability of almost submetries

Rong

2010

Front. Math. China

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“…Next we prove version 2 of Theorem 13, which relies on the following result found in [10]: On the other hand, if p j is different from p, then for each s, join a fixed minimal path in Σ from p j to p to the end of the path α s . Applying the argument above to this lengthened family of paths completes the proof.…”

Section: Then the Number Of Possibilities For The Isomorphism Class Omentioning

confidence: 97%

“…Aside from Wu's theorem, the most important ingredient in our proof of Theorem 2 will be a result of the author from [10] which compares the intrinsic distance function, d Fp , on a fiber (1) If B is simply connected, then there exists C depending only on {k,…”

Section: ) Vol(b) ≥ V Diam(b) ≤ D |Sec(b)| ≤ λ (2) Vol(m ) ≥ V Dmentioning

confidence: 99%

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Finiteness theorems for submersions and souls

Tapp

2001

Proc. Amer. Math. Soc.

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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. Counting Riemannian submersionsIn 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, π 1 : M 1 → B 1 and π 2 : M 2 → B 2 , to be C k -equivalent if there exists a C k diffeomorphism f : M 1 → M 2 which maps the fibers of π 1 C k -diffeomorphically onto the fibers of π 2 . Every Riemannian submersion is a fiber bundle (as long as the total space is complete), and C 0 -equivalence just means equivalence up to fiber bundle isomorphism.The following theorem was proven by the author:Then there are only finitely many C 1 -equivalence classes in the set of Riemannian submersions π : M n+k → B n for which: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 C 0 -finiteness) dropping the upper curvature bound on M :

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A parametrized compactness theorem under bounded Ricci curvature

2017

Front. Math. China

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Bounded Riemannian submersions

Cited by 12 publications

References 9 publications

Stability of almost submetries

Stability of almost submetries

Finiteness theorems for submersions and souls

A parametrized compactness theorem under bounded Ricci curvature

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