The main purpose of the present paper is to establish a theorem concerning the relation between the group of all projective transformations on an affinely connected manifold and the group of all affine transformations.

Let X be a space whose i-th homotopy group πi(X)vanishes for every i ≧ 0 except i = n ≧ 1, and whose n-th homotopy group is isomorphic to a group π. Then it is well known that the polyhedral homotopy type of X is completely determined by π and n. We call such a space a (π, n)-type space. Also it is well known that the minimal complex of the singular complex of a (π, n)-type space is isomorphic to the complex K(π, n)defined by S. Eilenberg and S. MacLane [1]. We know also that for any n ≧ 1 and any group π(abelian if n > 1) there exists a (π, n)-type space (See [6]).

In general a homogeneous space admits many invariant affine connections. Among these are certain connections which appear in many ways to be more natural than the others. We refer to the connections which K. Nomizu in [4] calls canonical affine connections of the first kind. When G is a compact connected Lie group and K a closed subgroup we called an invariant Riemannian metric on G/K, natural (in [2]) when it induced such a connection.

Mr. Z. Kuramochi has recently obtained the following very interesting result concerning the classification of Riemann surfaces [1]:

If we delete from a Riemann surface belonging to the class an arbitrary compact set, the resulting surface remains in the class .

In the present paper we indicate a new method of proving this theorem, simultaneously for the cases B and D.

No manifold had been known which can carry two distinct differentiable structures until the recent important contribution due to J. Milnor [7] concerning the 7-sphere appeared.

In connection with his work, there are several problems, for example, about the existence of any other manifold with such property, about the topological invariance of the Pontrjagin classes of manifolds, etc.; some of them will be discussed in the present note.

The questions concerning the dimension of the tensor product of two K-algebras have turned out to be surprisingly difficult. In this paper we follow a method using spectral sequences (§§1-3) which in some concrete cases yields complete results (§§4-5). In particular, complete results are obtained when r is a ring of matrices, triangular matrices, polynomials or rational functions, so that in the first three cases is respectively the ring of matrices, triangular matrices or polynomials with coefficients in the arbitrary algebra A.

Si X est un ensemble mesurable au sens de Lebesgue dans un espace euclidien En et Y est un ensemble mesurable au sens de Lebesgue dans un autre espace euclidien Em, alors X × Y est mesurable dans En+m= Enx Em et sa mesure est égale au produit de la mesure de X et celle de Y. Mais il n’existe pas de telle relation simple ni pour la capacité d’ordre général ni pour la mesure de Hausdorff de dimension générale. Dans le présent mémoire nous essayons d’évaluer la capacité des ensembles produits au moyen de la capacité ou de la mesure de Hausdorff des ensembles composants.

Many of the properties of analytic functions can be proved in a purely topological manner, so that such properties are then valid for the larger class of functions which are topologically equivalent to analytic functions. The importance of such functions has been recognized fairly recently, particularly in the theory of partial differential equations, where certain solutions have been shown to possess the topological properties of analytic functions, i.e., interiority and continuity, but not necessarily the analytic properties of complex differentiability and integrability.

On reviewing recently the proof which I gave for the Riemann mapping theorem for simply-connected Riemann surfaces several years ago [2], I observed that the argument which I used could be so modified that the assumption of a countable base could be completely eliminated. The problem of treating the Riemann mapping theorem without this assumption has been current for some time. The object of the present note is to give an account of a solution of this question. Of course, the classical theorem of Radó permits us to dispense with an attack on the Riemann mapping theorem which does not appeal to the countable base assumption. In this connection, we recall that Nevanlinna [4] has given a straightforward potential-theoretic treatment of the Radó theorem in which neither the Riemann mapping theorem (nor the notion of a universal covering) enters as they do in Radó’s proof. Nevertheless, a certain technical interest attaches to a direct treatment of the Riemann mapping theorem without the countable base assumption. An immediate byproduct of such a treatment is a simple proof of the Radó theorem which invokes the notion of a universal covering but in a manner different from that of Radó’s proof. Indeed, it suffices to note that a manifold has a countable base if the domain of a universal covering does.

In a recent paper [1], H. Ozeki has extended the author’s results [2] on local and infinitesimal holonomy groups for connections in linear fiber bundles (whose fiber is a vector space) to general fiber bundles whose structure group is a Lie group. Ozeki’s Lemma 7 (corresponding to the author’s Lemma 5.2) appears to be rather crucial in the development, while the proof is somewhat involved.—This note intends to present a more direct proof of the lemma, stating a case in which the local holonomy group H*(x)and the infinitesimal holonomy group H’(x)coincide:

Dans une Note parue dans ce Journal, M. Matsushima [1] a démontré le

Théorème. Dans un espace compact de Kähler-Einstein dont la courbure est positive, un vecteur analytique contrevariant u peut s’écrire d’une et d’une seule manière sous la forme u = v + Fw où v et w sont tous les deux les vecteurs de Killing et F le tenseur définissant la structure complexe de l’espace.

The most important thing for a set theory seems to be that it can generate new mathematical objects, and I think that there must be an underlying principle, simple and unique, which unifies the acts of generating. The naive set theory has a unique generating principle, which defines, by any proposition on a variable x, the set of all x’s satisfying the proposition. Certainly, we must restrict this generating principle so as to exclude all contradictions it contains, without losing its essential rôle as logic of mathematics, and at the same time we would like to keep its uniqueness and simplicity.

Some important notions in the theory of binary relations such as the relative product of two relations and the converse of a relation are defined in Whitehead and Russell’s “Principia mathematica” ([1]). McKinsey ([2]) and Tarski ([3]) gave their systems of postulates for the calculus of relations. Ore studied on equivalence relations ([4]) and Riguet on closures (fermeture) of relations ([5] and [6]), and they obtained remarkable results on the structure of these relations. The purpose of this paper is to examine the relative properties of some relations to each other, whose notions are closely related to the notions given by Riguet.

Let G be a finite group. A(left) G-module A of G is said to be of trivial cohomology when Hn(H, A)= 0 for all rational integers n and for all subgroups H of G. The main purpose of the present note is to determine the structure of finitely generated G-modules of trivial cohomology, which turns out to be remarkably simple (See Theorem 1 and Corollary 3 below). We prove also an (easy) localization theorem for cohomological triviality.

The aim of the present work is to determine the Galois group of the maximal abelian extension ΩA over an algebraic number field Ω of finite degree, which we fix once for all.

Let Z be a continuous character of the Galois group of ΩA/Ω. Then, by class field theory, the character Z is also regarded as a character of the idele group of Ω. We call such Z character of Ω. For our purpose, it suffices to determine the group Xl of the characters of Ω whose orders are powers of a prime number l.

Es ist wohlbekannt (Burnside [1]), dass(Sp1(GL(m, p)) < pm für ein belie-biges m und für eine von p(≠2)verschiedene ungerade Primzahl p1.

One of the most important problems in the potential theory is the one of capacitability, that is, whether the inner capacity of an arbitrary borelian subset B is equal to the outer capacity of B. As for the capacities induced by the Newtonian potentials and other classical potentials, Choquet [5] has shown that every borelian and, more generally, every analytic set are capacitable. He goes on as follows : first he shows that, for the Newtonian capacity f, the inequality of strong subadditivity holds, that is,

and then, using this inequality, he shows that the outer capacity f* has the analogous property to one of the outer measure, more precisely, if an increasing sequence {An} of arbitrary subsets converges to A, then f*(A) = limf*(An)- This property plays an important role in his proof.

Let Ω be an algebraic number field of finite degree, which we fix once for all, and let K be a cyclic extension over Ω such that the degree of K/Ω is a powerof a prime number l. It is obvious that the norm group NK/ΩeK of the unit group ek of K, being a subgroup of the unit group e of Ω contains the groupconsisting of all-th powersof ε∈e.

Using Frobenius automorphisms ingeniouslly, S. Lang has established an elegant theory of unramified class fields of function fields in several variables over finite fields [2]. As an application of class field theory and theory of reduction he has proved that any separable unramified abelian extension of a function field of one variable comes from a pull back of a separable ingeny of its jacobian variety [3].