340 research outputs found
Toponogov comparison theorem for open triangles
Dedicated to Professor Gromoll: The aim of our article is to generalize the
Toponogov comparison theorem to a complete Riemannian manifold with smooth
convex boundary. A geodesic triangle will be replaced by an open (geodesic)
triangle standing on the boundary of the manifold, and a model surface will be
replaced by the universal covering surface of a cylinder of revolution with
totally geodesic boundary. Applications of our theorem are found in our article
"Applications of Toponogov's comparison theorems for open triangles"
(arXiv:1102.4156).Comment: This version 4 (36 pages, no figures) is a version to appear in
Tohoku Mathematical Journa
Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. II
We prove, as our main theorem, the finiteness of topological type of a
complete open Riemannian manifold with a base point whose radial
curvature at is bounded from below by that of a non-compact model surface
of revolution 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 . Notice
that our model 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.Comment: 37 pages, No figures, Submitted to Transactions of Amer. Math. Soc.
(December 17, 2007), The [v1] is the 2nd revised version (October 17, 2008)
according to the 2nd referee's repor
Sufficient conditions for open manifolds to be diffeomorphic to Euclidean spaces
Let M be a complete non-compact connected Riemannian n-dimensional manifold.
We first prove that, for any fixed point p in M, the radial Ricci curvature of
M at p is bounded from below by the radial curvature function of some
non-compact n-dimensional model. Moreover, we then prove, without the pointed
Gromov-Hausdorff convergence theory, that, if model volume growth is
sufficiently close to 1, then M is diffeomorphic to Euclidean n-dimensional
space. Hence, our main theorem has various advantages of the Cheeger-Colding
diffeomorphism theorem via the Euclidean volume growth. Our main theorem also
contains a result of do Carmo and Changyu as a special case.Comment: This version 3 (13 pages, no figures) is a version to appear in
Differential Geometry and its Application
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