Approximating topological metrics by Riemannian metrics
Authors:
Steven C. Ferry and Boris L. Okun
Journal:
Proc. Amer. Math. Soc. 123 (1995), 18651872
MSC:
Primary 53C23; Secondary 57N60, 57R12
MathSciNet review:
1246524
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Abstract: We study the relation between (topological) inner metrics and Riemannian metrics on smoothable manifolds. We show that inner metrics on smoothable manifolds can be approximated by Riemannian metrics. More generally, if is a continuous surjection from a smooth manifold to a compact metric space with connected for every , then there is a metric d on X and a sequence of Riemannian metrics on M so that converges to (X, d) in GromovHausdorff space. This is used to obtain a (fixed) contractibility function and a sequence of Riemannian manifolds with as contractibility function so that is infinite dimensional. Using results of Dranishnikov and Ferry, this also gives examples of nonhomeomorphic manifolds M and N and a contractibility function so that for every there are Riemannian metrics and on M and N so that and have contractibility function and .
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 J. Dydak and J. J. Walsh, Infinite dimensional compacta having cohomological dimension two: An application of the Sullivan Conjecture, Topology 32 (1993), 93104. MR 1204409 (94b:55002)
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 M. Gromov, Large Riemannian manifolds, Lecture Notes in Math., vol. 1201, SpringerVerlag, Berlin and New York, 1986, pp. 108122. MR 859578 (87k:53091)
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 T. Moore, GromovHausdorff convergence to nonmanifolds, J. Geometric Anal. (to appear).
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 P. Peterson, V, A finiteness theorem for metric spaces, J. Differential Geom. 31 (1990), 387395. MR 1037407 (91d:53070)
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 S. Smale, A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. 8 (1957), 604610. MR 0087106 (19:302f)
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 J. Walsh, Isotoping mappings to open mappings, Trans. Amer. Math. Soc. 250 (1979), 121145. MR 530046 (80m:57008)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512465247
PII:
S 00029939(1995)12465247
Keywords:
Riemannian manifold,
length space,
celllike map
Article copyright:
© Copyright 1995
American Mathematical Society
