Zur Kennzeichnung Fanoscher Affin-Metrischer Geometrien

The underlying metric affine geometry, or metric projective geometry, can be recovered from Grassmann spaces associated with the family of regular subspaces of respective space. In other words, automorphisms of such Grassmann spaces are collineations witch preserve orthogonality of the respective underlying space. This generalizes results of Prażmowska et al. (Linear Algebra Appl 430:3066-3079, 2009) and Prażmowska andŻynel (Adv Geom, to appear). Mathematics Subject Classification (2010). 51F15, 51A45.A quadric Q in a projective space (or, in a more "abstract" and general language, a polar space embedded into a projective space) may determine quite various incidence geometries. One can consider the geometry on Q; the points and lines of this geometry are projective points and lines which lie on Q. This is a primary model of polar geometry. One can take Q and the lines which are intersections of Q and secants of Q (cf.[10]). One can consider as points the projective points not on Q, and as the lines projective lines which are tangent to Q (cf. [9]) or which are projective nonsingular (nonisotropic) or hyperbolic (cf. [12,13]).In this huge variety of incidence systems related to quadrics a very important subspaces are those radical free, called regular subspaces. Motivations to consider them come from reflection geometry, a quite natural language in which a geometry is characterized in terms of its admissible reflections (cf. [2,3,20]). Axes of such reflections are exactly those subspaces which we call regular. The concept of regular subspaces can be adopted in metric projective as well as in metric affine geometry and this is what we are doing here.

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“…e.g. [21]), and the metric projective space (P, ⊥) = Sub 1 (V), Sub 2 (V), ⊂, ⊥ with the conjugacy relation ⊥ defined on the points of P.…”

Section: Formulation Of Problemsmentioning

confidence: 99%

Grassmann spaces of regular subspaces

2010

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“…Their very close relationship has been hitherto overlooked. We first introduce an axiom system, consisting of A1-A8, for affine metric spaces of dimension ≥ 3, in the form presented in [25]. The language consists of one sort of variables, standing for points, and a ternary predicate L, with L(abc) to be read as 'points a, b, c are collinear (though not necessarily different)'.…”

Section: Introductionmentioning

confidence: 99%

“…Axiom A8 states that the dimension is at least 3, by stipulating that there are two skew lines a 1 a 2 and a 3 a 4 . We then introduce, following [25], axioms for a quaternary relation ≡, with ab ≡ cd to be read as 'segment ab is congruent to segment cd'. Axioms A9, A10, A11 ensure that ≡ is an equivalence relation and that all degenerate segments are congruent.…”

Section: Introductionmentioning

confidence: 99%

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Alexandrov–Zeeman type theorems expressed in terms of definability

Pambuccian

2007

Aequ. math.

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We formulate two remarkably elementary variants of theorems of the AlexandrovZeeman type, stated as definability statements, in minimalist axiomatic settings. The first is obtained by providing positive definitions for the notions of collinearity and segment congruence in terms of the notion of connectedness by a light ray, and is the logical counterpart of results stated in terms of mappings in [24] and [9]. The second is obtained by noticing that, in essence, results stating that mappings which preserve time-like lines preserve all lines, are the model-theoretic counterparts of definability results regarding the partial affine spaces introduced in [18]. (2000). 03C40, 03B30, 15A63, 51A15, 51F20, 83A05. Mathematics Subject Classification

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“…[3]) heigen zwei Geraden orthogonal, wenn sie sich schneiden und wenn sie orthogonale Richtungsvektoren besitzen. [3]) heigen zwei Geraden orthogonal, wenn sie sich schneiden und wenn sie orthogonale Richtungsvektoren besitzen.…”

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Bestimmung der orthogonalitätstreuen Permutationen euklidischer Räume

Benz

Schröder

1986

Geom Dedicata

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In einem fanoschen euklidischen Raum mit der Punktmenge P und der Geradenmenge G (vgl. [3]) heigen zwei Geraden orthogonal, wenn sie sich schneiden und wenn sie orthogonale Richtungsvektoren besitzen. Eine Permutation yon G heiBt orthooonalitdtstreu, wenn sie orthogonale Geraden in orthogonale Geraden/iberf/ihrt. W/ihrend man im ebenen Fall leicht orthogonalit/itstreue Permutationen von G angeben kann, die alles andere als Kollineationen sindman betrachte z.B. im ~2 diejenige Zuordnung, die die Geraden der Form (x,x) + •(1,0), (x,x) + ~(0, 1) ffir x~ R vertauscht und die alle iibrigen Geraden festl/if3t -liegt bei h6heren Dimensionen eine genau beschreibbare Situation vor.Wir werden zeigen, dab die orthogonalit/itstreuen Permutationen von G Automorphismen und damit insbesondere Kollineationen des betrachteten euklidischen Raumes sind, wenn dessen Dimension >14 ist. Im Falle der Dimension 3 erweisen sich die orthogonalit~itstreuen Permutationen yon G dagegen als Abbildungen, die man mit Hilfe yon Derivationen des Koordinatenk6rpers und yon Automorphismen des betrachteten euklidischen Raumes darstellen kann und die im allgemeinen keine Kollineationen sind.Ein ~ihnlich 'kontrolliertes' Ergebnis kann sich im ebenen Fall oftensichtlich nicht ergeben, und deshalb wird der ebene Fall im weiteren unberficksichtigt bleiben.Auf dem Wege zur L6sung des Problems stand uns H. Schaeffer in lebhaftem Gespr/ich mit mehreren hilfreichen Vorschl/igen zur Seite, und so bekunden wir ihm an dieser Stelle unseren herzlichen Dank. 1. FORMULIERUNGDERDARSTELLUNGSSATZE Gegeben sei ein Vektorraum V fiber einem kommutativen K6rper ~ mit 3 --< dim V -,< oo und mit char ~ ~ 2. Auf V existiere eine euklidische quadratische Form Q:V ~ IN mit der zugeh6rigen symmetrischen Bilinearform f, d.h. es gelte f(X,X)E~:=~\{O)VX~C~:=V\{O), wobei 0 der Nullvektor yon V ist. Im weiteren schreiben wir X-Y bzw. X 2 bzw. X _1_ Y statt f(X, Y) bzw. f(X, X) bzw. f(X, Y) = 0 ffir X, Y~ V. Zwei Elementeg:= A + KB undh:= C + ~D(A, CE V und B, DE V)der Geometriae Dedicata 21 (1986) 265-276. Qc) 1986 by D. Reidel Publishin 9 Company.

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Zur Kennzeichnung Fanoscher Affin-Metrischer Geometrien

Cited by 9 publications

References 5 publications

Grassmann spaces of regular subspaces

Grassmann spaces of regular subspaces

Alexandrov–Zeeman type theorems expressed in terms of definability

Bestimmung der orthogonalitätstreuen Permutationen euklidischer Räume

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