TY - JOUR

T1 - Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. II

AU - Kondo, Kei

AU - Tanaka, Minoru

PY - 2010/12

Y1 - 2010/12

N2 - 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.

AB - 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.

KW - Cut locus

KW - Geodesic

KW - Radial curvature

KW - Toponogov's comparison theorem

KW - Total curvature

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U2 - 10.1090/S0002-9947-2010-05031-7

DO - 10.1090/S0002-9947-2010-05031-7

M3 - Article

AN - SCOPUS:78650390327

SN - 0002-9947

VL - 362

SP - 6293

EP - 6324

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 12

ER -