In this note, we treat the problem to determine the conformal structure of the closed surface by the structure of the differentiable function algebra as the normed algebra with a certain norm.

A similar investigation is found in Myers [1]. He concerns himself with determining the Riemannian structure of the compact manifold using a certain normed algebra of differentiable functions.

In his paper [1] F. Kasch developed a theory of Frobenius extensions as a generalization of the theory of Frobenius algebras. In it he established a very interesting relationship between the Frobenius property of an extension and that of its endomorphism ring [1, Satz 5], from which he further derived the Frobenius extension property of Galois extensions of simple rings [1, Satz 6]; with these results he kindly responded to what had been vaguely “conjectured” (as he wrote) by one of the writers on the connection between Galois theory and the theory of Frobenius algebras [2].

In this paper we study two features of relative homological algebra of Frobenius extensions, the one is the relative dimension and the other is the relative complete resolution. The theory of Frobenius extensions was developed by F. Kasch [4] as a generalization of Frobenius algebras. We deal with only Frobenius extensions of finite rank (see §2 for the definition).

Let M be a connected metric space and H an isometry group of M which leaves fixed a point p in M. M is said H isotropic at p when, for any two points q and r of M at the same distance from p, there exists an isometry in H which carries q to r. When H coincides with the maximum isometry group leaving p fixed, M is said merely isotropic at p.

The purpose of this paper is to find out what can be learned about valuation rings, and more generally Prufer rings, from a study of their injective modules. The concept of an almost maximal valuation ring can be reformulated as a valuation ring such that the images of its quotient field are injective. The integral domains with this latter property are found to be the Prufer rings with a (possibly) weakened form of linear precompactness for their quotient fields.

It is well known that any distributive lattice can be imbedded in a Boolean algebra ([1], [2], [4] and others). This imbedding is in general only finitely isomorphic in the sense that the imbedding preserves finite sums (supremums) and finite products (infimums) (but not necessarily infinite ones). Indeed, in order to be able to be imbedded into a Boolean algebra completely isomorphically (i.e. preserving every supremum and infimum) a distributive lattice L must satisfy the infinite distributive law, as the infinite distributivity holds in Boolean algebras. The main purpose of this paper is to prove that the converse is also true, that is, any infinitely distributive lattice can be imbedded completely isomorphically in a Boolean algebra (Theorem 6). Since we show, on the other hand, that any relatively complemented distribu tive lattice is infinitely distributive (Theorem 2), Theorem 6 implies that every relatively complemented distributive lattice can be imbedded completely isomorphically in a Boolean algebra (Theorem 4).

Dans ce travail nous étudierons le problème de classifier les espaces fibres principaux holomorphes de groupe GL(m, C) sur un tore complexe admettant des connexions holomorphes i.e. la classification des espaces fibres vectoriels holomorphes sur un tore complexe admettant des connexions holomorphes. Récemment M. Matsushima [6] a démontré qu’un espace fibré vectoriel holomorphe sur un tore complexe admettant une connexion holomorphe de fibre de dimension 2 possède nécessairement une connexion holomorphe intégrable c.à.d. une connexion holomorphe dont la forme de courbure est nulle. D’autre part d’après Atiyah [1] on sait qu’un espace fibré holomorphe de base M et de groupe structural G admettant une connexion holomorphe intégrable est uu espace fibré principal associé au revêtement universel de la base M par une représentation du groupe fondamental de M dans G et vice versa.

Dans la première partie de cet article, on généralise une construction donnée dans [2], conduisant à l’homologie des espaces classifiants de groupe Γ. Cette construction utilise au départ un espace fibre principal Y de groupe Γ, le cas traité dans [2] étant celui où Y = Γ. Elle permet de définir dans le module de cohomologie de la base X de Y une filtration par une suite de sous-modules dont l’intersection est le module des classes caractéristiques. Dans la seconde partie, la filtration précédente est étendue à Hl(X, G) où G est le faisceau des germes d’applications continues de X dans un groupe topologique G.

Le but principal de ce mémoire est d’étudier les espaces fibres principaux différentiables et holomorphes de groupe abelien connexe ayant pour base une C-variété au sens de Wang [12]. Une C-variété X étant une variété complexe compacte simplement connexe et homogène, X est la base d’un espace fibre principal qui est un groupe de Lie où opère un sous-groupe fermé par les translations à droite comme groupe structural. Pour cette raison, on considère d’abord dans le § 1 les espaces fibres principaux différentiables de groupe abelien connexe A ayant pour base la base X d’un certain espace fibre principal differentiate Y de groupe B. Parmi ces espaces fibres principaux les plus simples sont ceux qui sont associés à Y par les homomorphismes du groupe B dans le groupe A.

Guldin-Pappus’s theorem about the volume of a solid of rotation in the euclidean space has been generalized in two ways. G. Koenigs [1] and J. Hadamard [2] proved that the volume generated by a 1-parametric motion of a surface D bounded by a closed curve c is equal to where are quantities attached to D with respect to a rectangular coordinate system, while are quantities determined by our motion.

In this paper we investigate indices of umbilics of a closed surface in the euclidean space. Most part of the discussion is concerned with a symmetric tensor field of degree 2, or rather a direction field, on a Riemannian manifold of dimension 2.

The local theory of continuous (infinite) pseudo-groups of transformations was originated by S. Lie, and developed by himself, F. Engel, E. Vessiot, E. Cartan, etc. In the beginning, the definition was not clear and we can find several different definitions in the papers of pioneers. In 1902, E. Cartan introduced a definition using his theory of exterior differential systems and made an extensive study in his series of papers [1], [2], and [31 The writer will adopt his definition in this series of papers. A continuous pseudo-group of transformations is, roughly speaking, a collection of real (or complex) analytic homeo-morphisms of domains in a real (or complex) euclidean space, which is closed under the operations of composition and inverse, and which forms the general solutions of a system of partial differential equations.

In the theory of meromorphic functions, it is important to investigate the properties of covering surfaces generated by their inverse functions. For this purpose, the study of properties of a non-compact region of a Riemann surface is useful.