Showing papers in "Journal of Geometry in 1981"

Zur Theorie Linearer Abbildungen I

Abstract: Given an n-dimensional pappian projective space, any linear mapping in the grassmannian of d-dimensional subspaces induces a linear mapping in the associated Grassmann-variety which is the restriction of one and only one linear mapping in the projective space generated by the variety.

Extensions of distance reducing mappings to piecewise congruent mappings on ℓm

Abstract: In this note I will show any distance reducing mapping f: M → ln, where M is a finite subset of lm (m ≤ n), can be extended to a piecewise conqruent mapping f: lm → ln.

A Tverberg-type generalization of the Helly number of a convexity space

Abstract: In 1966 H. Tverberg gave a far reaching generalization of the well-known classical theorem of J. Radon. In this paper a similar generalization of the classical Helly theorem is given and it is shown that among these two generalized theorems a relationship holds similar to a theorem proved by F.W. Levi in 1951. Also the generalized Helly theorem in the convex product and convex sum space are investigated.

A characterisation of the embeddings of PG(m,q) into PG(n,qr).

Abstract: Let Sm be an embedding of PG(m,q) into PG(n,qr),with n < m, such that Sm generates PG(n,qr). Sm can be obtained as a projection from a (m−n−1)-dimensional subspace Vm−n−1 into a non incident n-dimensional subspace Vn of some strong embedding S of PG(m,q) into PG(m,qr).

Transversal designs associated with frobenius groups

Abstract: To any Frobenius group G (of degree s, with Frobenius complement of order k) we associate an (s,k) -transversal design Δ(G) which admits G as a point-regular collineation group. Δ(G) is in fact also a dual translation net and furthermore admits a flag-regular collineation group. Also, Δ(G) has two orthogonal resolutions. Conversely, we will characterize the Frobenius groups among the point-regular collineation groups of resolvable transversal designs. We also exhibit two further classes of flag-regular transversal designs. Finally, we completely determine the possible parameters of TD's constructed as above.

A new partial geometry with parameters $(s,t,\alpha)=(7,8,4)$

Abstract: A partial geometry with parameters as given in the title is constructed by use of the 240 points closest to the origin in the lattice E8.

Zur Kennzeichnung Fanoscher Affin-Metrischer Geometrien

Abstract: 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.

Espaces polaires faiblement plonges dans un espace projectif

Abstract: The classification of polar spaces which are fully embedded in a finite-dimensional projective space, is now achieved (see for instance [8]). The present paper initiates the study of polar spaces embedded in a larger sense, by defining a weak embedding of them.

Elementary abelian 2-groups on the line at infinity of translation planes

Abstract: Let G be a subgroup of the linear translation complement of a translation plane of order qd with kernel GF(q) and let ¯G be the factor group modulo the scalars. We show that if ¯G is elementary abelian of order 2a, and if each involution in ¯G has a conjugate class of length greater than a+1 then 2e divides d, where e=[1/2(a+1)]−1. We show that one of Walker's planes is a counterexample if we drop the condition on lengths of conjugate classes. The Walker plane in question turns out to be of rank 3. This is one of Walker's planes of order 25 and was not previously known to have rank 3.

There exist at least three non-isomorphic S(2,4,28)'S

Abstract: A Steiner system S(2,4,28) is constructed, which is non-isomorphic to any of the known S(2,4,28)'s.

Visibility and parallelotopes in projective space

Abstract: It is shown that if γ is a finite set of points in an ordered d-dimensional projective space, d≥2 and if all the hyperplanes spanned by points of γ are considered, then every such a hyperplane is visible from at most 2d distinct points of γ; if a hyperplane bounds 2d distinct residences, then it contains exactly d points of γ. These results have been proved previously for d=2 and d=3.

Ordinals and scattered compact ordered spaces

Abstract: Scattered Compact Ordered Spaces (SCOS) are studied with respect to their well ordered images and preimages.

Zur Isomorphie vektorieller Gruppen und affiner Liniengeometrien

Abstract: Aus der Isomorphie zweier vektorieller Gruppen G I und G 2 (siehe [I]) folgt natfirlich die Isomorphie die zugeh6rigen Gruppen G I und G 2 und auch die Isomorphie der abgeleiteten affinen Liniengeometrien A(GI) und ~(G2) (siehe []]); die Umkehrungen gelten nicht, da es einerseits Gruppenmit wesentlich verschiedenen vektoriellenAbbildungen gibt, andererseits gibt es affine Liniengeometrien, die durch nicht isomorphe vektorielle Gruppen beschrieben werden k6nnen ([I]). Es blieb die Frage offen, ob bei einer Gruppe G mit den vektoriellenAbbildungen S I und S 2 aus der Isomorphie der beiden Geometrien A(G, $I) und A(G, $2) schon die Isomorphie der vektoriellen Gruppen (G, $I) und (G, $2) folgt. Dies aber ist zu verneinen, wie in dieser Note gezeigt wird. Bei der Konstruktion yon Beispielen werden teilweise kategorielle Begriffe verwandt, weswegen diese zun~chst kurz erl~utert werden. FQr die fundamentalen Begriffsbildungen in der Kategorientheorie sei auf [2] verwiesen.

Zur Darstellung der Klassischen Gruppen durch Clifford-Algebren

Abstract: Let (V,K,Q) be a finite dimensional metric vector space, and let C(Q) be the corresponding Clifford algebra. In this paper, representations of the classical groups On(V,Q) and POn(V,Q) by C(Q) are given for the case of arbitrary rank and arbitrary characteristic. For example, the following is proved: If R is the domain of regularity of C(Q) with center ZR then — apart from three exceptions — R/ZR is isomorphic to On(V,Q), and ancorresponding result is obtained for POn (V,Q).

On the Thomsen condition for webs

Abstract: By finding the single identity exactly corresponding to Thomsen's condition, a simpler proof is given of the fact that a web satisfies that condition if and only if the associated quasigroup is isotopic to an abelian group.

Algebraisierung Scharf 2-Fach Transitiver Permutationsmengen durch Quasibereiche

Abstract: Sharply 2-transitive permutation sets (M,Γ) with the property: every α e Γ which interchanges two distinct elements of M is in\(\mathfrak{J}\) and\(\mathfrak{J}\)Γ VΓ\(\mathfrak{J}\)⊂Γ (where\(\mathfrak{J}\):= {ω e Γ; ω2=id ≠ ω}) can be characterized algebraically by quasi-domains as defined by G. Kist.

Riemannian manifolds with a parallel field of complex planes

Abstract: The purpose of this paper is to discuss Riemannian manifolds which admit a parallel field of complex planes, consisting of vectors of the form\(a + \sqrt { - 1} b\), where a,b are real orthogonal vectors of equal length. Using the Nirenberg Frobenius Theorem [12], it follows that these are reducible Riemannian manifolds, whose metric is locally a sum of a Kahler and of a Riemann metric, and we are calling thempartially Kahler manifolds.

Über gemeinsame Lote Windschiefer geraden in elliptischen Räumen

Abstract: Let the following motions in a projective elliptic space (K,π) be called “normal forms”:Subspace-reflections, rotations, the product of a rotation and a point-reflection whose center lies on the axis of the rotation and the product of two rotations, whose axes of rotation are conjugated.

Torsen Konstanten Flächenwinkels

Abstract: In this paper we consider certain developable surfaces, which are circumscribed to a given ruled surface. We give new characterisations of helicoidal ruled surfaces and a result concerning the lines of curvature of ruled surfaces.

Zentrale und axiale Kollineationen in projektiven Hjelmslev-Ebenen

Abstract: Es gibt in projektiven Hjelmslev-Ebenen zentrale (axiale) Kollineationen, die keine Achse (Zentrum) haben. Das Produkt zweier zentraler Kollineationen s und t mit gemeinsamer Achse kann eine axiale Kollineation ohne Zentrum sein oder auch eine axiale Kollineation mit einem Zentrum, das auf keiner Verbindungsgeraden der Zentren von s und t liegt.

A function of direction associated with n congruences of an orthogonal ennuple in a Riemannian hypersurface

Abstract: In a hypersurface Vn belonging to a Riemannian space Vn+1 we choose n congruences of an orthogonal ennuple of unit vectors of contravariant components λ h¦ i (h=1,2,...,n) and denote, by ωkk the normal curvature of the hypersurface in the direction of the unit vector of components λ k¦ i . In the present paper, we have shown that the expression $$\frac{\partial }{{\partial s_k }}\omega _{kk} - 2\mathop \Sigma \limits_{h = 1}^n \omega _{kh} \gamma _{hkk}$$ is a function of direction, where the symbol ∂/∂sk indicates the differentiation in the direction of the vector of components λ k¦ i and that ωkh (h≠k) and γlhk (l,h,k=1,2,...,n) are, respectively, the invariants of the geodesic torsion of the curve of the congruence with unit tangent vector of components λ k¦ i and Ricci's coefficients of rotation of the orthogonal ennuple.

Maximal subgroups of the orthogonal group

Abstract: Let V be an n-dimensional regular quadratic space over a field K of characteristic not 2. Assume n ≥ 4. Let W be a regular hyperplane and v a nonzero vector orthogonal to W. Suppose every regular hyperplane in W is universal. If σ is an isometry of V not leaving W invariant, then σ, together with the isometries of W, generate the orthogonal group of V, with one exception.

Quadriques partielles d'indice deux

Abstract: L'origine de ce travail reside dans l'observation que le groupe de Higman-Sims possede une geometrie tres proche de celle d'une quadrique d'indice de Witt deux, constituee de 100 points, de droites de 2 points et de cercles de 6 points. Notre but est de decrire un systeme d'axiomes qui caracterise simultanement la geometrie des droites et des cercles des quadriques finies d'indice deux et la “quadrique” de Higman-Sims.