Publisher Summary This chapter discusses linear topological geometries and offers some observations concerning the role of topology in the foundations of geometry. Continuity certainly is an essential part of the geometric imagination. In fact, topological elements are indispensable for characterizations of the classical geometries. This was clear at the turn of the century, despite the fact that the abstract concepts and notions of topology had not yet emerged. Thus, Hilbert expressed the necessary continuity assumptions in terms of ordering properties. One direction of research in the first decades of this century was constituted by the control (and, to some extent, the elimination) of the influence of ordering axioms. The notion of a topological projective plane was introduced and studied systematically by Skornjakov, Salzmann and Freudenthal. Unlike Hilbert's system of axioms, this topological approach includes complex and quaternion projective spaces, as well as the octonion plane; compare the lucid review in Freudenthal. There is a related approach, which eliminates all incidence axioms in flavor of assumptions on topological transformation groups characterizing the groups of motions.

Linear Topological Geometries

In this chapter we will discuss kinematic algebras and their applications in geometry. We begin (§1) with a survey of the development of kinematics. The subjects covered in this paper will be found in §2.

Kinematic Algebras and Their Geometries

Let V be a vector space over the commutative field K such that char K
 2 ∧ 2 ≤ dim V ≤ ∞, and let Q:V → K be a quadratic form of rank ≥ 2. The pair (A(V,K),ξQ), consisting of the affine space A(V,K) and the congruence relation ξQ, defined by (a,b)ξQ (c,d) ⇔ Q(a−b) = Q(c−d) ∀(a,b),(c,d) ∃ V×V, is called an affine-metric fano-space of rank ≥ 2. In this paper, such spaces are characterized by three simple geometrical properties.

Zur Kennzeichnung Fanoscher Affin-Metrischer Geometrien

Let $V$ be a valuation domain with quotient field $K$. Given a pseudo-convergent sequence $E$ in $K$, we study two constructions associating to $E$ a valuation domain of $K(X)$ lying over $V$, especially when $V$ has rank one. The first one has been introduced by Ostrowski, the second one more recently by Loper and Werner. We describe the main properties of these valuation domains, and we give a notion of equivalence on the set of pseudo-convergent sequences of $K$ characterizing when the associated valuation domains are equal. Then, we analyze the topological properties of the Zariski-Riemann spaces formed by these valuation domains.

The Zariski–Riemann space of valuation domains associated to pseudo-convergent sequences

Publisher Summary
This chapter discusses metric concepts such as distance, congruence, orthogonality, angle, circle, quadric, reﬂection, motion, similarity, and the corresponding theorems. Linear algebras are generalized to vector spaces that have infinite dimension or characteristic 2. If U is a vector subspace of V, by the axiom of choice, there exists a basis B of U, and its cardinal number |B| is the dimension of U, noted dim U. Moreover, there exists a vector subspace W of V such that U Ө W = V and it turns out that W is isomorphic to the quotient space V/U, which implies dim W = dim V/U. The subspaces U and W are called complementary in V, and dim V/U is called the co-dimension of U in V. By definition, the vector subspace U is a line (resp., a plane, a hyperplane) of V provided dim U = 1 (resp., dim U = 2, dim V/U = 1).

Chapter 17 – Metric Geometry

Im ersten Abschnitt werden metrische affine Raume mit Hilfe eines Axiomen-systems eingefuhrt, das nur Inzidenz- und Kongruenzforderungen enthalt und der von E Schroder in [13] angegebenen Beschreibung verallgemeinerter euklidischer Raume ahnelt Jeder endlich-dimensionale Vektorraum mit einer nicht-ausgearteten quadratischen Form uber einem kommutativen Korper erweist sich als metrisch in unserem Sinne ((12)) Im zweiten Teil wird gezeigt, das sich die kongruenzerhaltenden Permutationen eines solchen Vektorraums algebraisch bequem darstellen lassen, wenn die quadratische Form kompositionserhaltend ist, dh wenn eine Kompositionsalgebra vorliegt Diejenigen unter ihnen, die 0 und 1 festlassen, sind Jordanautomorphismen und semilineare Abbildungen zugleich ((26) und (27)) Dieses Resultat wird im letzten Abschnitt zur Beschreibung der Isometrien von Kompositionsalgebren herangezogen

Kongruenzerhaltende Permutationen von Kompositionsalgebren

It is the aim of this paper to present an extensive class of disconnected locally compact nearfields. They are all Dickson nearfields and derived from tamely ramified extensions of local fields.

Konstruktion lokalkompakter Fastkörper

It is well known that there are three types of prolongations of valuations from a field K to $K(t)$, t a transcendent over K. In this paper, we give a representation of the equivalence and isomorphy classes of each type with the help of an algebraic closure $\overline{K}$ of K. Furthermore, upper and lower bounds for the cardinalities of these classes are presented, and we answer the question regarding when a subfield of $\overline{K}$ allows a classification of the prolongation of valuations to $K(t)$.

Aquivalenz und Isomorphie bei Bewertungsfortsetzungen auf einfach-transzendente Erweiterungen

Two problems are treated: (A) We look for all couplings of a finite cyclic group G. (B) We try to determine all finite groups G bearing couplings with cyclic derivations. These problems will be solved completely, if G is a p-group and nilpotent, respectively. The results enable us to construct couplings with cyclic Dickson groups.

Heinz Wähling

Papers

Kongruenzerhaltende Permutationen von Kompositionsalgebren

Konstruktion lokalkompakter Fastkörper

Aquivalenz und Isomorphie bei Bewertungsfortsetzungen auf einfach-transzendente Erweiterungen

Gruppenkopplungen Mit Zyklischer Ausgangsgruppe Bzv7. Zyklischer Ableitung

ApproximationssÄtze FÜr Topologische, Bewertete und Angeordnete FastkÖrper