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$ C^k$-quasi-isometry sets are pre-compact

Author(s): F. T. Farrell; P. Ontaneda
Journal: Proc. Amer. Math. Soc. 138 (2010), 737-741.
MSC (2000): Primary 58A05, 58D05, 58D17, 58D19
Posted: October 9, 2009
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Abstract: Let $ M$ be a closed smooth manifold. By an argument formally similar to one used in constructing the Levi-Civita connection, it is shown that $ C^k$-quasi-isometry sets in $ DIFF^{k+1}(M)$ are $ C^{k+1}$-bounded, where $ 0\leq k< \infty$. This implies, using the Arsela-Ascoli theorem, that such sets are pre-compact in $ DIFF^{k}(M)$.

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Additional Information:

F. T. Farrell
Affiliation: Department of Mathematics, State University of New York, Binghamton, New York 13902

P. Ontaneda
Affiliation: Department of Mathematics, State University of New York, Binghamton, New York 13902

DOI: 10.1090/S0002-9939-09-10132-6
PII: S 0002-9939(09)10132-6
Received by editor(s): March 4, 2009,
Received by editor(s) in revised form: March 6, 2009
Posted: October 9, 2009
Additional Notes: Both authors were partially supported by NSF grants.
Communicated by: Chuu-Lian Terng
Copyright of article: Copyright 2009, American Mathematical Society

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