Kei Kondo

Bio: Kei Kondo is an academic researcher from Tokai University. The author has contributed to research in topics: Curvature & Riemannian manifold. The author has an hindex of 6, co-authored 26 publications receiving 177 citations. Previous affiliations of Kei Kondo include Okayama University & Yamaguchi University.

Generalized space forms

Abstract: 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 RP n 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.

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below: I

Abstract: 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 π. 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.

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

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.

Topology of Complete Manifolds with Radial Curvature Bounded from Below

Abstract: 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.

Toponogov comparison theorem for open triangles

Abstract: 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.

Measure and Integral

Abstract: Mass und Integral und ihre Algebraisierung Von Prof C Caratheodory Herausgegeben von P Finsler, A Rosenthal und R Steuerwald (Lehrbucher und Monographien aus dem Gebiete der Exakten Wissenschaften Mathematische Reihe, Band 10) Pp 337 (Basel und Stuttgart: Birkhauser Verlag, 1956) 3850 francs; 3850 DM

The cut locus of a two-sphere of revolution and Toponogov's comparison theorem

Abstract: We determine the structure of the cut locus of a class of two-spheres of revolution, which includes all ellipsoids of revolution. Furthermore, we show that a subclass of this class gives a new model surface for Toponogov's comparison theorem.

Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds

Abstract: Given a manifold $$M$$ , we build two spherically symmetric model manifolds based on the maximum and the minimum of its curvatures. We then show that the first Dirichlet eigenvalue of the Laplace–Beltrami operator on a geodesic disk of the original manifold can be bounded from above and below by the first eigenvalue on geodesic disks with the same radius on the model manifolds. These results may be seen as extensions of Cheng’s eigenvalue comparison theorems, where the model constant curvature manifolds have been replaced by more general spherically symmetric manifolds. To prove this, we extend Rauch’s and Bishop’s comparison theorems to this setting.

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below: I

Abstract: 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 π. 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.