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

Kei Kondo, Minoru Tanaka

Research output: Contribution to journalArticlepeer-review

14 Citations (Scopus)

Abstract

We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold M with a base point p ∈ M whose radial curvature at p is bounded from below by that of a non-compact model surface of revolution 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 p ∈ M. Notice that our model 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.

Original languageEnglish
Pages (from-to)6293-6324
Number of pages32
JournalTransactions of the American Mathematical Society
Volume362
Issue number12
DOIs
Publication statusPublished - Dec 2010
Externally publishedYes

Keywords

  • Cut locus
  • Geodesic
  • Radial curvature
  • Toponogov's comparison theorem
  • Total curvature

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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