Kei Kondo

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 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 Milnors open conjecture on the fundamental group of complete open manifolds.

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

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.

Incorporation of cationic electron donor of Ni-pyridyltetrathiafulvalene with anionic electron acceptor of polyoxometalate

A new salt-[Ni(II)(DMSO)(5)(TTFPy)](2)[α-SiW(12)O(40)] (1)-based on polyoxometalates was prepared by coordinating a cationic electron donor of pyridyltetrathiafulvalene (TTFPy) with Ni(II). Although the TTFPy molecule did not form a salt with the anionic α-[SiW(VI)(12)O(40)](4-) because of the weak charge-transfer (CT) interaction, the coordination of Ni with the pyridyl moiety permitted salt formation driven by electrostatic interaction, giving a single crystal of 1. Crystallographic analysis, UV-vis and IR spectroscopy and electrochemical characterization revealed that the fully oxidized α-[SiW(VI)(12)O(40)](4-) was crystallized with the neutral TTFPy moiety from the acetonitrile solution because of the low electron-withdrawing ability of α-[SiW(VI)(12)O(40)](4-), forming a brown-orange crystal. The crystal colour quickly turned to black by immersing in methanol, due to CT from TTF moiety to α-[SiW(VI)(12)O(40)](4-), which was caused by the solvent effect. Increase in the solvent acceptor number from 18.9 for acetonitrile to 41.3 for methanol resulted in the enhancement of the electron withdrawing ability of α-[SiW(VI)(12)O(40)](4-) by 0.317 V in methanol.

Grove-Shiohama type sphere theorem in Finsler geometry

AbstractFrom radial curvature geometry’s standpoint, we prove a few sphere theoremsof the Grove-Shiohama type for certain classes of compact Finsler manifolds. 1 Introduction Beyond a doubt, one of the most beautiful theorems in global Riemannian geometry isthe diameter sphere theorem of Grove and Shiohama [GS]. In their proof, Toponogov’scomparison theorem (TCT) was very first applied seriously together with the critical pointtheory, introduced by themselves, of distance functions. That is, if a complete Riemannianmanifold X has a critical point, say q ∈ X \ {p} , of the distance function d p to a point p ∈ X , then q is the cut point of p . And hence d p is not differentiable at q . However, theyovercame the analytical obstruction by applying the original TCT to the triangle △ ( pxy )with the interior angle ∠( pxy ) ≤ π/ 2 at x . That is the point, i.e., they took the manifoldinto their hands by directly drawing segments on it.Our purpose of this article is to prove a sphere theorem of the Grove-Shiohama typefor a certain class of forward complete Finsler manifolds whose radial flag curvatures arebounded below by

The topology of an open manifold with radial curvature bounded from below by a model surface with finite total curvature and examples of model surfaces

We will construct 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 will prove that a complete non-compact Riemannian manifold M is homeomorphic to the interior of a compact manifold with boundary, if the manifold M is not less curved than a non-compact model surface of revolution, and if the total curvature of the model surface is finite and less than

Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below, III

2\pi

TOPOLOGY OF COMPLETE FINSLER MANIFOLDS WITH RADIAL FLAG CURVATURE BOUNDED BELOW

. Hence, in the first result mentioned above, we may treat a much wider class of metrics than that of a complete non-compact Riemannian manifold whose sectional curvature is bounded from below by a constant.

Sufficient conditions for open manifolds to be diffeomorphic to Euclidean spaces

We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold

Topology of Complete Manifolds with Radial Curvature Bounded from Below

M

A Toponogov type triangle comparison theorem in Finsler geometry ⁄y

with a base point