The invariant affine connection over a coset space G/J of a Lie group G have been discussed by various authors. Recently, Nomizu [8] gave a systematic study of this problem when J is reductible in G. Among other results, he established a 1-1 correspondence between the invariant affine connections and certain multilinear mappings, and calculated the torsion and curvature. For canonical affine connection of the second kind, the holonomy group was also given.

In this Part (III) the proofs in UL are given for fundamental formulas concerning the following dependent variables:

• Elementary set {a1, … , an};

• Ordered pair ‹a, b›;

• Image by σ of aσ‘a;

• Image by σ of elements of aσ“a;

• Domain of operator Dσ;

• Range of operator Wσ;

• Uniqueness Un;

• Bi-uniqueness Un2;

• Inverse operator σ-1;

• One-to-one mapping Map2a, b;

• Composition of operators σ o τ;

• Restriction of operator

• Identical mapping ℓ

Let X(A, ω), ω∈Ω, be Wiener process on the probability space (Ω, B, P) depending on a point A of an N dimensional Euclidean space EN.

The purpose of this paper is to prove the following

Theorem. Let M be a Riemannian manifold of dimension n and let ξ be a Killing vector field (i.e., infinitesimal isometry) of M. Let F be the set of points x of M where ξ vanishes and let F = ∪ Vi, where the Vi’s are the connected components of F. Then (assuming F to be non-empty)

Let us consider a surface S of class C3 in Euclidean space E3 and

(1.1) xi = xi(u1, u2) (i, j, k = 1, 2, 3)

be its parametric representation of class C3.

Let J be a directed set and let {Λj, φij} be a direct system of rings indexed by J and with limit Λ. Let {Aj, φij} be a direct system of groups indexed by J. Assume that each Aj is a left Λj-module and that φij(λa) = φij(λ) φij(a) for each λ ∈ Λj, a ∈ Aj. Then the limit A of (Aj, φij) is a left Λ-module.

A ring R is said to be right (left) hereditary if every right (left) ideal in R is projective, that is, a direct summand of a free R-module. Cartan and Eilenberg [3, p. 15] ask whether there exists a right hereditary ring which is not left hereditary. The answer: yes.

Let R be a ring with identity element 1, and let S be a subring of R containing 1. We consider R-modules on which 1 acts as the identity map, and we shall simultaneously regard such R-modules as S-modules in the natural way. In [4], we have defined the relative analogues of the functors of Cartan-Eilenberg [1], and we have briefly treated the corresponding relative analogues of module dimension and global ring dimension.

In the present note, using the results in the previous paper, we shall prove the following existence theorem:

THEOREM. Let k be a finite field with q elements and K/k be a regular extension of dimension one over k. Then, if q ≧ 11 and the genus gK of K/k is greater than one, there exists an unramified separable infinite solvable extension of K ivhich is regular over k.

In der vorliegenden Arbeit beschäftigen wir uns mit dem Problem der geschlossenen Geodätischen auf geschlossenen Mannigfaltigkeiten. Dieses Problem geht bekanntlich auf Morse ([4], [5]) zurück. Wir machen aber von seinem Kunstgriff des “zyklischen” Produktes [4] keinen Gebrauch; vielmehr kommt es uns hier darauf an, den Funktionalraum unseres Variations-problems direkt in Betracht zu ziehen, wie es Morse beim analogen Problem der geodätischen Linien mit festen Randpunkte herausgearbeitet hat. Es wird sich im folgenden zeigen, wie sich sein Verfahren auf den Fall der geschlossenen Geodätischen übertragen lässt.

Let A be a quasi-Frobenius algebra over a field K. A has a complete (co)homology theory which may be established upon an augmented acyclic projective complex, i.e. a commutative diagram

of A-double-modules with exact horizontal row, projective Xp, and with epimorphic resp. monomorphic ε and t. Negative-dimensional cohomology groups, over an A-double-module, are expected to be in close relationship with (ordinary positive-dimensional) homology groups.

In § 1 the usage and conventions which are used in the deductions in UL are explained. In §§ 2-4 some metatheorems concerning the deductions are proved. Namely, in § 2 the order of proof constituents in a proof is investigated; in § 3 the applicability of composite proof constituents are proved on the basis of § 2; in §4 the place of the “ordinarily used” principle of extensionality in a proof is specified. In §5 a sufficient condition of the mechanization of mathematics is given in such a manner that the mechanical and non-mechanical parts in solving mathematical problems are separated in accord with the usual way of thinking in mathematics.

In the papers [11] and [18] Rohlin and Thom have introduced an equivalence relation into the set of compact orientable (not necessarily connected) differentiable manifolds, which, roughly speaking, is described in the following manner: two differentiable manifolds are equivalent (cobordantes), when they together form the boundary of a bounded differentiable manifold. The equivalence classes can be added and multiplied in a natural way and form a graded algebra Ω relative to the addition, the multiplication and the dimension of manifolds. The precise structures of the groups of cobordism Ωk of dimension k are not known thoroughly. Thom [18] has determined the free part of Ω and also calculated explicitly Ωk for 0 ≦ k ≦ 7.

Dans ce mémoir nous démontrons que le groupe d’automorphismes d’un espace fibre principal analytique complexe à base compacte est un groupe de Lie complexe par la topologie de la convergence compacte et ensuite nous étudierons quelques relations entre le groupe d’automorphismes d’un espace fibre principal analytique complexe et le groupe d’homéomorphisme analytiques de sa base.

Ein gewöhnliches Verfahren bei dem Studium der Riemannschen Flächen besteht darin, dass man die für die komplexe Ebene benutzten Methoden auf allgemeine Riemannsche Flächen überführt und die bekannten Satze fur schlichtartige Gebiete auf den Riemannschen Flachen verallgemeinert. Dises Verfahren schlägt zuweilen fehl, da manche Sätze, welche im Falle der schlichtartigen Gebiete richtig sind, im Falle beliebiger Riemannscher Flächen nicht mehr wahr bleiben.