Showing papers in "Kodai Mathematical Journal in 1990"
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TL;DR: In this article, the authors studied the Dirichlet energy in the three-dimensional case of contact three-manifolds and introduced the torsion τ =LχQg, namely the Lie derivative of g with respect to X0.
Abstract: where <3tt(ώ) denotes the space of all associated Riemannian metrics to the contact form ω. This functional was studied by Blair and Ledger [2] in general dimension. However the three-dimensional case has many special features to merit a separate study. Chern and Hamilton [7] introduced the torsion τ=LχQg, namely the Lie derivative of g with respect to X0, in their study of compact contact three-manifolds, and studied the Dirichlet energy
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TL;DR: In this article, it was shown that the rank of the Z2-cohomology ring of a compact proper Dupin hypersurface has a constant value 1/2, where ρ is the cross-ratio of the four points on a projective line corresponding to the four curvature spheres of the hypersurfaces.
Abstract: A hypersurface M in a standard sphere S is said to be Dupin if each of its principal curvatures is constant along its corresponding curvature surfaces. If the number of distinct principal curvatures is constant, then M is called a proper Dupin hypersurface. There is a close relationship between the class of compact proper Dupin hypersurfaces and the class of isoparametric hypersurfaces. Miinzner [11] showed that the number g of distinct principal curvatures of an isoparametric hypersurface must be 1, 2, 3, 4 or 6. Thorbergsson [15] then showed that the same restriction holds for a compact proper Dupin hypersurface embedded in S by reducing that case to a situation where Mίmzner's argument can be applied. This also implied that the rank of the Z2-cohomology ring in both cases must be 2g. Later Grove and Halperin [6] found more topological similarities between these two classes of hypersurfaces. All of this led to the conjecture [5, p. 184] that every compact proper Dupin hypersurface in S is equivalent by a Lie sphere transformation to an isoparametric hypersurface. The conjecture was known to be true in the cases g=l (umbilic hypersurfaces), g=2[4] and g=3[7]. Recently, however, counterexamples to the conjecture for g=4 have been discovered by Miyaoka and Ozawa [10] and by Pinkall and Thorbergsson [14]. Miyaoka and Ozawa also produced counterexamples in the case £—6. In all cases, the proof that the counterexamples are not Lie equivalent to an isoparametric hypersurface uses the so-called Lie curvature ^introduced by Miyaoka [8]. For a proper Dupin hypersurface Mwith four principal curvatures, Ψ is the cross-ratio of these principal curvatures. Viewed in the context of Lie sphere geometry, Ψ is the cross-ratio of the four points on a projective line corresponding to the four curvature spheres of M. Hence, Ψ is a natural Lie invariant. From Mϋnzner's work, it is easy to compute that Ψ has a constant value 1/2 on a Dupin hypersurface which is Lie equivalent to an isoparametric hypersurface. In projective geometric terms, this means that the four curvature spheres at each point of M form a harmonic set. For the counterexamples to the conjecture above, Ψ does not have the constant value 1/2.
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TL;DR: In this article, smooth functions were used for pseudo-immersion in the Taylor series and the filter was used as a pseudo-imersion filter. Reference CTN-ARTICLE-1990-001
Abstract: Keywords: smooth functions ; pseudo-immersion ; Taylor series ; filter ; matrix Reference CTN-ARTICLE-1990-001doi:10.2996/kmj/1138039222 Record created on 2008-12-12, modified on 2016-08-08
TL;DR: In this article, it was shown that for n ≥ 5, there exists a constant δ(n) with 1/4 < δ (n) < 1/ε(n)-pinched sectional curvatures such that any weakly stable Yang-Mills connection over a simple connected compact Riemannian manifold M of dimension n with δ n-pinched sectionsal curvature is always flat.
Abstract: It is proved that for n≥5 there exists a constant δ(n) with 1/4<δ(n)<1 such that any weakly stable Yang-Mills connection over a simple connected compact Riemannian manifold M of dimension n with δ(n)-pinched sectional curvatures is always flat. The pinching constants are possible to compute by elementary functions. Moreover we give some remarks on stability of Yang-Mills connections over certain symmetric spaces
TL;DR: In this article, it was shown that every component subgroup of a torsion-free Kleinian group is also finitely generated, and the minimal number of generators of component subgroups is quantitatively estimated.
Abstract: Kulkarni and Shalen [6] explained topologically Ahlfors' ίiniteness theorem and its two major supplements, the area-inequalities [2] and the finiteness of the cusps [16], considering the core of the quotient 3-manifold of finitely generated torsion-free Kleinian groups. As a consequence, they obtained the sharp inequalities on the number of cusps. In this article we treat chiefly another result from the finiteness theorem. It is that every component subgroup of a finitely generated Kleinian group is also finitely generated. This assertion is qualitative. So we will estimate quantitatively the minimal number of generators of component subgroups. Our consequence (§2) is;