Showing papers in "Tohoku Mathematical Journal in 1958"

On the differential geometry of tangent bundles of riemannian manifolds ii

Abstract: H.Poincare used the tangent sphere-bundles of ovaloids in three dimensional Euclidean space, i.e. the phase spaces of the ovaloids, to prove the existence of certain closed geodesies on the ovaloids. He introduced a Riemannian metric on the tangent sphere-bundles and considered the geodesic flow on it. As the metric of tangent bundles of Riemannian manifolds seems to be important, we would like to study differential geometry of tangent bundles of Riemannian manifolds by introducing on it natural Riemannian metrics. In this papar we shall do it by restricting ourselves only to the tangent bundles T{M).

On the projection of norm one in w*-algebras, iii

Abstract: This paper is a continuation of the author's preceding papers [8], [9], in which we discuss certain existence-problems of σ-weakly continuous projections of norm one of different types of W*-algebras. By a projection of norm one we mean a projection mapping from a Banach space onto its subspace whose norm is one. In the following we concern with the projection of norm one in a W*-algebra M. We denote by Mjc the space of all σ-weakly continuous linear functionals on M. On the other hand M* means the conjugate space of M and the second conjugate space of M is written by M** usually. However, in case M is a W*-algebra M** is the W*-algebra that plays a special role for M (cf. [3], [7]) so that we denote especially by M. A positive linear functional φ on a W*-algebra is called singular if there exists no non-zero positive σ-weakly continuous functional such as ψ S φ. The closed subspace of M* generated by all singular linear functionals is denoted by MJ. Then we get M* = M* © M* : the sum is /^direct sum. A uniformly continuous linear mapping TΓ from a W*-algebra M to another W*-algebra N is called singular if MN*) c Mί where τr means the transpose of TΓ. All other notations and definitions are referred to [7] and [8]. Before going to discussions, the author expresses his hearty thanks to Mr. M. Takesaki for his valuable suggestions and co-operations. 1. General decomposition theorem.

Absolute summability of Rademacher series

Abstract: is convergent, o being the n th Cesaro mean of order a of the series an. The summability IC, a1 is the ordinary absolute summability IC, aI. Among the many theorems on the absolute summability, one of the most interesting is the so-called high-indices theorem. By the Zygmund highindices theorem [5], if the series an is lacunary, that is, its terms are all zero except for the terms with indices nu=01 (j=1, 2.....), and if the series is summable IAI, then it turns out to be absolutely convergent or summable IC, 01. Flett [2] studied an extension of this result to the summability IAI and gave an inequality corresponding to that of the Zygmund theorem [5]. But he has left open the

On Euclidean connections in a Finsler manifold

Abstract: Introduction. In a general Finsler space S. S. Chern [2] defined an infinite number of Euclidean connections which include the connection defined by E. Cartan and others as a particular cases. The purpose of this note is to discuss in a modern view-point Euclidean connections in a Finsler manifold and to give a gee netrical interpretation to the connections defined by S. S. Chern. Throughout the whole discussion the following conventions are adopted: By differentiability we understand always that of class C, and differential forms of degree 1 of class C are all called as 1-forms. Given a 1-form over a manifold M by w, we denote the restriction of it at a point x of M by wx. We denote wx by w, too for brevity if it is not ambiguous from the context. For vector field and so on this convention is also applied. Let us assume that Latin indices b, c, d, h, i, j, k, t run from 1 to n and Creek indices a, B, y, b from 1 to -1. The auther wishes to express here his sincere gratitude to Prof. S. Sasaki for his kind assistance during the preparation of the manuscript.

On the realization of the stiefel-whitney characteristic classes by submanifolds

Abstract: INTRODUCTION Chapter I. REALIZATION OF THE STIEFEL-WHITNEY CLASSES BY INDUCED SCHUBERT MANIFOLDS 1. Preliminaries 2. Subvarieties Corresponding to Wi 3. Examples 4. A Sufficient Condition II. REALIZATION OF THE STIEFEL-WHITNEY CLASSES BY THE CONSTRUCTION OF MAPPINGS 5. A Necessary Condition 6. On The Spaces K(Z2,2; Z,A; k ) and M(O(2)) 7. W> in VQ III. SPECIAL MANIFOLDS 8. Totally Realizable Manifolds 9. Complete Intersections of Hypersurfaces REFERENCES