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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. II


Authors: Kei Kondo and Minoru Tanaka
Journal: Trans. Amer. Math. Soc. 362 (2010), 6293-6324
MSC (2010): Primary 53C21; Secondary 53C22
Published electronically: July 13, 2010
MathSciNet review: 2678975
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Abstract: We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold $ M$ with a base point $ p \in M$ whose radial curvature at $ p$ is bounded from below by that of a non-compact model surface of revolution $ \widetilde{M}$ which admits a finite total curvature and has no pair of cut points in a sector. Here a sector is, by definition, a domain cut off by two meridians emanating from the base point $ \tilde{p} \in \widetilde{M}$. Notice that our model $ \widetilde{M}$ does not always satisfy the diameter growth condition introduced by Abresch and Gromoll. In order to prove the main theorem, we need a new type of the Toponogov comparison theorem. As an application of the main theorem, we present a partial answer to Milnor's open conjecture on the fundamental group of complete open manifolds.


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Additional Information

Kei Kondo
Affiliation: Department of Mathematics, Tokai University, Hiratsuka City, Kanagawa Pref. 259–1292 Japan
Email: keikondo@keyaki.cc.u-tokai.ac.jp

Minoru Tanaka
Affiliation: Department of Mathematics, Tokai University, Hiratsuka City, Kanagawa Pref. 259–1292 Japan
Email: m-tanaka@sm.u-tokai.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05031-7
PII: S 0002-9947(2010)05031-7
Keywords: Toponogov’s comparison theorem, cut locus, geodesic, radial curvature, total curvature
Received by editor(s): December 17, 2007
Received by editor(s) in revised form: June 30, 2008
Published electronically: July 13, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.