Showing papers in "Journal of Geometry in 1993"
24 citations
21 citations
TL;DR: In this paper, it was shown that the compact cyclides of Dupin are of infinite type and this result provides further support for a conjecture of Chen, which is known as the infinite compact cyclide conjecture.
Abstract: We show that the compact cyclides of Dupin are of infinite type; this result provides further support for a conjecture of Chen.
15 citations
TL;DR: In this article, a generalization of Chen submanifolds is defined and a characterization for k-th-Chen-submanifold is given. But this characterization is restricted to the case of k-submansions.
Abstract: As a generalization of Chen submanifolds,k-th Chen submanifolds are defined. A characterization for them is proved. Spherical 2nd Chen submanifolds are discussed. For a compact submanifoldM with parallel second fundamental form it is proved thatM is ak-th Chen submanifold if and only ifM is ofk-type.
9 citations
TL;DR: In this article, the shape operator of the mean curvature vector is the tangent part of a fixed linear transformation of the Euclidean space ℝm, and a hypersurface of restricted type is either minimal, a part of the product of a sphere and a linear subspace or a cylinder on a plane curve of restricted types.
Abstract: A submanifoldM
n
of the Euclidean space ℝm is said to be of restricted type if the shape operator of the mean curvature vector is the tangent part of a fixed linear transformation of ℝm. We show that a hypersurface of restricted type is either minimal, a part of the product of a sphere and a linear subspace or a cylinder on a plane curve of restricted type. Finally we classify plane curves of restricted type.
9 citations
9 citations
9 citations
TL;DR: In this paper, the Darboux rotation for space curves in Euclidean space E3 is decomposed into two simultaneous rotations, and the axes of these simultaneous rotation are joined by a simple mechanism.
Abstract: The Darboux rotation for space curves in Euclidean space E3 is decomposed into two simultaneous rotations. The axes of these simultaneous rotations are joined by a simple mechanism. One of these axes is a parallel of the principal normal of the curve, the direction of the other is the direction of the Darboux vector of the curve. This decomposition of the Darboux rotation yields a necessary condition for the curve to be closed.
8 citations
TL;DR: In this article, it was shown that any non-degenerate polar space of rank at least 4 with at least 3 points on each line is isomorphic to one of the classical examples.
Abstract: We show that any nondegenerate polar space of rank at least 4 with at least 3 points on each line is isomorphic to one of the classical examples. Although this is not a new result, the authors have recently found a simple proof which works equally well for spaces of finite or infinite rank. Here we present an alternate approach to part of that classification.
7 citations
TL;DR: In this paper, a completely different approach was used to determine all-ruled surfaces in Euclidean space ℝ3, which are (at least locally) WEINGARTEN-surfaces under theminimal assumption Φ∈C2.
Abstract: Referring to articles of BELTRAMI (1865), DINI (1866) and CHARIAR (1978), but using a completely different approach, we determine allruled surfaces Φ in Euclidean space ℝ3, which are (at leastlocally) WEINGARTEN —-surfaces under theminimal assumption Φ∈C2. Theskew ruled WEINGARTEN —surfaces can be characterized by havingconstant invariants d ≠ O (parameter of distribution), k (skewness of distribution) and σ (striction angle); theirfunctional (WEINGARTEN-)relation between the mean curvature H and the Gaussian curvature K of Φ is of the form H=α (-K)1/4 +β (-K)3/4 with arbitrary real constants α,β. These facts allow various geometric interpretations.
6 citations
TL;DR: In this paper, rational curves and surfaces on quadric surfaces are considered as solutions of certain diophantic equations in polynomial rings and a representation formula from number theory gives rise to a generalization of stereographic projection.
Abstract: The paper presents a powerful construction of rational curves and surfaces on quadric surfaces. These curves and surfaces are considered as solutions of certain diophantic equations in polynomial rings. A representation formula from number theory gives rise to a generalization of stereographic projection. The paper discusses the properties of this map. Some connections to advanced geometry and to the foundations of geometry are outlined.
TL;DR: In this article, a Bruck loop on ℝ×ℝ is constructed using the function λ(x)=cosh x, and the automorphisms δa,b of the K-loop are investigated.
Abstract: Using the function λ(x)=cosh x, K-loops on ℝ×ℝ are constructed. Since every K-loop is a Bruck loop, we have also examples for Bruck loops. Furthermore we investigate the group of the automorphisms δa,b of the K-loop which satisfy the equation a⊕(b⊕c)=(a⊕b)⊕δa,b(c).
TL;DR: In this paper, the notion of generalized elliptic spaces was extended to generalized euclidean spaces by Kroll and Sorensen [S1] and a one-to-one correspondence between these structures and quaternion fields was established.
Abstract: The notion of an elliptic plane given 1975 by K. Sorensen [S1] will be extended to the notion of a “generalized elliptic space”. Each such elliptic space is derivable from a generalized euclidean space in the sense of H.-J. Kroll and K. Sorensen [KS]. For the case that the euclidean resp. elliptic space has the dimension 3 resp. 2 there is a one to one correspondence between these structures and quaternion fields. Each quaternion field of characteristic ≠ 2 defines in a natural way a 4-dimensional euclidean and a 3-dimensional elliptic space. But, in general, we do not obtain in this way all 4- resp. 3-dimensional geometries. The geometries derivable from quaternion fields will be characterized. Both of these two classes of geometries are provided with different structures, so that there are different automorphism groups, which will be studied.
TL;DR: The Pasch criterion is proved andrapezium and Parallelogram Configurations in the theory of oriented parallelity are considered and the Pasch theorem is proved.
Abstract: Trapezium and Parallelogram Configurations in the theory of oriented parallelity are considered. The Pasch theorem is proved.
TL;DR: One approach to the ancient problem of "duplicating the cube" is related to a peculiar extremal problem in Euclidean and non-Euclidean planes as discussed by the authors.
Abstract: One approach to the ancient problem of “duplicating the cube” is related to a peculiar extremal problem in Euclidean and non-Euclidean planes. The non-Euclidean instances of this problem lead to the study of a family of octavic curves in the real projective plane.
TL;DR: In this paper, the transformation process introduced in [8] was generalized to find the affine Andre planes and the nonplanar nearfield discovered by H.Karzel and W.Kerby.
Abstract: In this note we generalize the transformation process introduced in [8]. This generalization allows to find the nonplanar nearfield discovered by H.Karzel and W.Kerby, [5], [6], and the affine Andre planes.
TL;DR: In this paper, the question under which conditions an arbitrary permutation ϕ of a quadric satisfying X ≈Y ⇔ Xϕ ≈ Yϕ ∀ X,Y e F can be extended to a collineation of the given space is answered.
Abstract: Two pointsX, Y of a quadricF in an affine or projective space are called 0-distant, writtenX ≈Y, if they coincide or if their connecting line is a subset ofF. In this paper, answers are given to the question under which conditions an arbitrary permutation ϕ ofF satisfyingX ≈Y ⇔Xϕ ≈Yϕ ∀X,Y e F can be extended to a collineation of the given space.
TL;DR: In this paper, the definition of a PK-nearring is given and it is shown that every PK nearring is a local ternary ring, and further algebraic properties of the corresponding PK-planes are related to geometrical ones of corresponding nearrings.
Abstract: In the first section the definition of a PK-nearring is given and it is shown that every PK-nearring is a local ternary ring. Then a PK-nearring is used to construct PK-planes and further algebraic properties of PK-nearrings are related to geometrical ones of the corresponding PK-planes. Finally, a wide class of PK-nearrings is presented by applying the Dickson-process to the ring of formal power series. Throughout the following all nearrings shall be left-distributive. For the notation and basic properties of nearrings the reader is referred to Pilz [5].
TL;DR: In this article, a uniform generalization of the inflection circle of plane Euclidean kinematics to CK-motions of 1st kind, which turns out to be a curve of 2nd order, is presented.
Abstract: In [12] the local kinematics of all seven Cayley/Klein-planes (CK-planes) was developed in a largely uniform way. The motions of Euclidean, pseudo-Euclidean, elliptic and hyperbolic planes were called CK-motions of 1st kind. They are fundamentally different from quasielliptic, quasihyperbolic and isotropic motions (CK motions of 2nd kind). In this paper we consider a uniform generalization of the inflection circle of plane Euclidean kinematics to CK-motions of 1st kind, which turns out to be a curve of 2nd order. It can be shown that many important properties of the Euclidean inflection circle are retained. We also generalize the Euclidean cuspidal circle, inflection point and cuspidal point.
TL;DR: In this article, the authors generalize the precedent study of two particular cubics, related to any given triangle, by the construction of analagmatic class six cubics under a common quadratic involutive transformation referred again to a given triangle.
Abstract: We generalize the precedent study of two particular cubics, related to any given triangle, by the construction of analagmatic class six cubics under a common quadratic involutive transformation referred again to any given triangle. As resume, we point out some insights for future extensions to generalized theories about the cubics.
TL;DR: In this paper, the class of almost constant-type manifolds is introduced and some theorems concerning Ricci tensors, scalar curvatures, bisectional curvatures and curvature identities are proved.
Abstract: We introduce the class of almost constant-type manifolds and prove some theorems concerning Ricci tensors, scalar curvatures, bisectional curvatures and curvature identities. The above class is also studied in relation to other known classes of almost hermitian manifolds.
TL;DR: This paper studies the geometric continuity of arbitrary order between tensor product Bézier surfaces of arbitrary degree in Computer Aided Geometric Design.
Abstract: Geometric continuity is a central topic of Computer Aided Geometric Design. In this paper we study the geometric continuity of arbitrary order between tensor product Bezier surfaces of arbitrary degree.
TL;DR: In this paper, the authors proved and disproved the following conjecture of A. Bezdek and K. Kannan: s2(n) = s1([n/3]).
Abstract: Let s1 (n) denote the largest possible minimal distance amongn distinct points on the unit sphere\(\mathbb{S}^2\). In general, let sk(n) denote the supremum of thek-th minimal distance. In this paper we prove and disprove the following conjecture of A. Bezdek and K. Bezdek: s2(n) = s1([n/3]). This equality holds forn > n0 however s2(12) > s1(4).
TL;DR: In this paper, Hohenberg's axonometric views of an object of the physical space are constructed from two auxiliary views in the drawing plane "without auxiliary lines" without auxiliary lines.
Abstract: F. Hohenberg proposed a method to construct axonometric views of an object of the physical space from two auxiliary axonometric views in the drawing plane “without auxiliary lines”. This elegant method is unjustifiedly neglected in common courses of descriptive geometry and engineering graphics, although it is a wellsuited tool to visualize even higher dimensional objects and one can easily adapt it to computer aided descriptive geometry. This paper discusses obvious generalizations of Hohenberg's method and its relationship to other geometric mappings.
TL;DR: In this article, the authors clarify the combinatorial nature of projective coordinate systems of modular upper continuous lattices and generalize the classical relationship between 3-dimensional Desarguesian configurations and coordinate systems for projective 3-spaces.
Abstract: This note clarifies the combinatorial nature of projective coordinate systems of modular upper continuous lattices. It generalizes the classical relationship between 3-dimensional Desarguesian configurations and coordinate systems of projective 3-spaces.
TL;DR: Bichara and Somma as mentioned in this paper showed that a Schubert graph is isomorphic to the graph representing the flags of a Boolean lattice, where the lines have exactly two points.
Abstract: We introduce the concept of Schubert graphs, as Schubert spaces — in the meaning of A. Bichara and C. Somma [2] — whose lines have exactly two points. They turn out to be isomorphic to particular Cayley graphs of symmetric groups; this leads also to a new proof of a well-known characterization of symmetric groups. In connection with [2], we prove that a Schubert graph is isomorphic to the graph representing the flags of a Boolean lattice. Finally, we discuss the independence of the axioms.
TL;DR: In this paper, it is shown that the intersection points of anyd pairwise orthogonal supporting hyperplanes lie on a fixed sphere, which is a quantitative improvement of the uniqueness theorem in the form of a stability result.
Abstract: It is known that the ellipsoids ind-dimensional Euclidean space, ford ≥ 3, are characterized among all convex bodies by the property that the intersection points of anyd pairwise orthogonal supporting hyperplanes lie on a fixed sphere. The subject of the present note is a quantitative improvement of this uniqueness theorem in the form of a stability result.
TL;DR: The study of combinatorial handles in (n+1)-coloured graphs with boundary was studied in this article, where it was shown that cancelling of a handle from an n+1-coloured graph can cancel the associated complex.
Abstract: In this work we complete the study ofcombinatorial handles in (n+1)-coloured graphs with boundary, introduced in [G1], [L] and [GV] for graphs with empty boundary and in [BGV] for 3-coloured graphs with boundary. In particular, we study the cancelling of a combinatorial handle from an (n+1)-coloured graph and its effects on the associated complex.
TL;DR: In this article, a complete solution of the existence problem of handcuffed designs H(v,3,1) having an oval was given. And for each admissible v>5, a H (v, 3, 1) without ovals was constructed.
Abstract: In this paper we give a complete solution of the existence problem of handcuffed designs H(v,3,1) having an oval. Moreover we construct for each admissible v>5 a H(v,3,1) without ovals.