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Showing papers in "Tohoku Mathematical Journal in 1969"




Journal ArticleDOI
TL;DR: This paper shall define a conformal Killing tensor in another way and generalize some results about a conformAL Killing vector to the conformalkilling tensor.
Abstract: where pc is a certain vector field. Because we can easily show that a conformal Killing tensor in this sense is a Killing tensor, i.e., we have pc = 0. Thus this definition of conformal Killing tensor is meaningless. In this paper we shall define a conformal Killing tensor in another way and generalize some results about a conformal Killing vector to the conformal Killing tensor. The definition which we shall adopt is suggested by the following fact. A parallel vector field in the Euclidean space E induces a

167 citations


Journal ArticleDOI
TL;DR: In this article, the authors define a Sasakian manifold with pseudo-Riemannian metric, and discuss the classification of the manifold with constant φ-sectional curvatures, and prove that such a manifold is of constant curvature.
Abstract: Introduction. Sasakian manifold with Riemannian metric is defined by S. Sasaki [5]. In this paper, we want to define Sasakian manifold with pseudo-Riemannian metric, and discuss the classification of Sasakian manifolds. In section 1, we define a Sasakian manifold (with pseudo-Riemannian metric). In section 2, we define the model spaces of Sasakian manifolds which are used in section 4 for the classification of Sasakian manifolds of constant φ-sectional curvatures. In section 3, we discuss Z)-homothetic deformation which is defined by S. Tanno [9], and prove some fundamental lemmas concerning completeness of the deformed metric. In section 5, we prove that a Sasakian manifold, satisfying R(X, Y) R = 0 for all tangent vectors X and Y, is of constant curvature. In section 6, we discuss a Sasakian manifold M^ which is properly and isometrically immersed in £f.

139 citations




Journal ArticleDOI
TL;DR: In this paper, a tensor field for the study of hypersurfaces in almost complex manifolds was introduced and some theorems on normality of induced almost contact structures in the language of tensor fields were discussed.
Abstract: Introduction. The present author [6] has shown that an orientable hypersurface in an almost complex manifold can be given an induced almost contact structure, and studied conditions for the induced structure to be normal. On the other hand, K.Yano and S.Kobayashi [8] and K.Yano and S.Ishihara [9] have introduced the notions of vertical, complete and horizontal lifts of tensor fields and connections to tangent bundles. Above all, if the base manifold is almost complex, then the complete or horizontal lift of the structure defines an almost complex structure in the tangent bundle. It is also known, cf. [5], that the tangent bundle of a Riemannian manifold is given an almost Kahlerian structure.1) Basing on these two kinds of results, we shall be able to induce various kinds of almost contact structures into tangent sphere bundles. In this paper, we shall consider a class of hypersurfaces with some property in tangent bundles ; the class contains the tangent sphere bundle of a manifold. In §§1 to 3, we shall introduce a tensor field for the study of hypersurfaces in almost complex manifolds and state some theorems on normality of induced almost contact structures in the language of the tensor field. In • ̃• ̃4 to 8, various kinds of almost contact structures will be induced to hypersurfaces in tangent bundles and the normality of the structures will be discussed. In • ̃8, we shall show that, given the almost Kahlerien structure in the tangent bundle of a Riemannian manifold, the induced almost contact structure of the tangent unit-sphere bundule is K-contact if and only if the base manifold is of positive constant curvature.

63 citations







Journal ArticleDOI
TL;DR: In this paper, a spectral theory for closed linear operators on a Banach space is presented, which is based on the theory of decomposable operators with functional calculus on their spectrum.
Abstract: In this paper we shall construct a certain spectral theoryfor closed linear operators on a Banach space.These operators have a suitable spectral behaviour on subsets of theirspectra but we must eliminate some residual part which do not offer informationabout the intimate structure of the considered objects, at least from our pointof view.It will be easy to see that this theory contains many examples of operators,bounded or not, having a functional calculus on their spectrum [1], [2], [3], [6],[8], [9].A permanent model for our construction will be the theory of decomposableoperators on a Banach space [7], [2].Throughout this paper the sets of points will be taken in CΌo^Cuf







Journal ArticleDOI
Kôji Uchida1
TL;DR: In this article, the authors generalize some duality theorems concerning Galois cohomology groups of finite modules to the case of finitely generated modules (see also [6]).
Abstract: 1. Notations and Summary of Results. J.Tate [7] has proved duality theorems concerning Galois cohomology groups of finite modules . We generalize some of these theorems to the case of finitely generated modules (See also [6]). Tate has also stated about strict cohomological dimension . Our first aim was to prove this statement which seems to have important meanings in the number theory. But we have not succeeded yet. Let G be a profinite group. We call A a G-module only when it is equal to the union of the submodules A, where H runs over the open subgroups of G and H operates trivially on AH. Cohomology groups of such modules in positive dimensions are well known. Cohomology groups in negative and zero dimensions are introduced by Poitou1). By definition