Language：English   Publishing type：Research paper (scientific journal)   Publisher：KYUSHU UNIV, FAC MATHEMATICS  

We recently established a Toponogov-type triangle comparison theorem for a certain class of Finsler manifolds whose radial flag curvatures are bounded below by that of a von Mangoldt surface of revolution. In this article, as its applications, we prove the finiteness of topological type and a diffeomorphism theorem to Euclidean spaces.

DOI： 10.2206/kyushujm.68.347

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：Osaka University and Osaka City University, Departments of Mathematics  

Recently we generalized Toponogov's comparison theorem to a complete Riemannian manifold with smooth convex boundary, where a geodesic triangle was replaced by an open (geodesic) triangle standing on the boundary of the manifold, and a model surface was replaced by the universal covering surface of a cylinder of revolution with totally geodesic boundary. The aim of this article is to prove splitting theorems of two types as an application. Moreover, we establish a weaker version of our Toponogov comparison theorem for open triangles, because the weaker version is quite enough to prove one of the splitting theorems.&0a;

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Other Link： http://hdl.handle.net/11094/25094

Language：English   Publishing type：Research paper (scientific journal)   Publisher：DUKE UNIV PRESS  

We construct distinctive surfaces of revolution with finite total curvature whose Gauss curvatures are not bounded. Such a surface of revolution is employed as a reference surface of comparison theorems in radial curvature geometry. Moreover, we prove that a complete noncompact Riemannian manifold M is homeomorphic to the interior of a compact manifold with boundary if the manifold M is not less curved than a noncompact model surface (M) over tilde of revolution and if the total curvature of the model surface (M) over tilde is finite and less than 2 pi. By the first result mentioned above, the second result covers a much wider class of manifolds than that of complete noncompact Riemannian manifolds whose sectional curvatures are bounded from below by a constant.

DOI： 10.1215/00277630-1959451

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：MATH SOC JAPAN  

This article is the third in a series of our investigation on a complete non-compact connected Riemannian manifold M. In the first series [KT1], we showed that all Busemann functions on an M which is not less curved than a von Mangoldt surface of revolution (M) over tilde are exhaustions, if the total curvature of (M) over tilde is greater than pi. A von Mangoldt surface of revolution is, by definition, a complete surface of revolution homeomorphic to R-2 whose Gaussian curvature is non-increasing along each meridian. Our purpose of this series is to generalize the main theorem in [KT1] to an M which is not less curved than a more general surface of revolution.

DOI： 10.2969/jmsj/06410185

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER HEIDELBERG  

We investigate the finiteness structure of a complete non-compact n-dimensional Riemannian manifold M whose radial curvature at a base point of M is bounded from below by that of a non-compact von Mangoldt surface of revolution with its total curvature greater than pi. We show, as our main theorem, that all Busemann functions on M are exhaustions, and that there exists a compact subset of M such that the compact set contains all critical points for any Busemann function on M. As corollaries by the main theorem, M has finite topological type, and the isometry group of M is compact.

DOI： 10.1007/s00208-010-0593-4

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：TOHOKU UNIVERSITY  

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.

DOI： 10.2748/tmj/1318338947

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p is an element of 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. (C) 2011 Elsevier B.V. All rights reserved.

DOI： 10.1016/j.difgeo.2011.04.040

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold M with a base point p is an element of M whose radial curvature at, p is bounded from below by that of a non-compaci model surface or revolution (M) over tilde 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 oil by two meridians emanating from the base point p is an element of (M) over tilde. Notice that our model (M) over tilde does not always satisfy the diameter growth condition introduced by A bresch and Gromoll. In order to prove the main theorem, we need it 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 sloop of complete open manifolds.

DOI： 10.1090/S0002-9947-2010-05031-7

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Language：English   Publishing type：Research paper (scientific journal)  

We show a radius sphere theorem for a compact Riemannian manifold whose radial curvature at the base point is bounded from below by that of a 2-sphere of revolution. The diameter sphere theorem is expanded to a wide class of metrics. © 2007 International Academic Printing Co. Ltd. All rights reserved.

DOI： 10.3836/tjm/1202136689

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：BIRKHAUSER VERLAG AG  

We investigate the topology of a complete Riemannian manifold whose radial curvature at the base point is bounded from below by that of a von Mangoldt surface of revolution. Sphere theorem is generalized to a wide class of metrics, and it is proven that such a manifold of a noncompact type has finitely many ends.

DOI： 10.1007/s00039-007-0625-8

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Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.3836/tjm/1244208683

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Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

Spaces with radially symmetric curvature at base point p are shown to be diffeomorphic to space forms. Furthermore, they are either isometric to R-n or S-n under a radially symmetric metric, to RPn with Riemannian universal covering of S-n equipped with a radially symmetric metric, or else have constant curvature outside a metric ball of radius equal to the injectivity radius at p.

DOI： 10.1090/S0002-9947-02-02966-5

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