Showing papers in "Geometriae Dedicata in 1987"

Isometries of the unit sphere

Abstract: It is shown that isometries between the unit spheres of finite dimensional Banach spaces necessarily map antipodal points to antipodal points.

Curvatures and currents for unions of sets with positive reach

Abstract: A class of subsets of ℝd which can berepresented as locally finite unions of sets with positive reach isconsidered. It plays a role in PDE's on manifolds with singularities.For such a set, the unit normal cycle (determining the d − 1curvature measures) is introduced as a (d − 1)-currentsupported by the unit normal bundle and its properties are established.It is shown that, under mild additional assumptions, the unit normalcycle (and, hence, also the curvature measures) of such a set can beapproximated by that of a close parallel body or, alternatively, by themirror image of that of the closure of the complement of the parallelbody (which has positive reach). Finally, the mixed curvature measuresof two sets of this class are introduced and a translative integralgeometric formula for curvature measures is proved.

On tilings of the plane

Abstract: The paper discusses homeomorphic types of (periodic) tilings of the plane in terms of their associated Delaney symbol. Such a symbol consists of a (finite) set D on which three involutions σ0, σ1 and σ2 act from the right such that σ0σ2=σ2σ0 and there are two maps m 01, m 12 : D satisfying certain compatibility conditions. It is shown how the barycentric subdivision of a tiling can be used to define its Delaney symbol and that the symbol characterizes the tiling up to (equivariant) homeomorphisms. Furthermore, it is shown how properties of the tiling can be recognized from corresponding properties of the symbol and how this technique can be used to enumerate various types of tilings with specific properties. If necessary, this enumeration can be done by appropriate computer programs. Among other results, we have been able to vindicate the results by Grunbaum et al., announced in [8]. Finally, some recursive enumeration formulas, based on the Delaney symbol technique, are stated.

The combinatorics of colored triangulations of manifolds

Abstract: Foundations for the topic of crystallizations are proposed through the more general concept of colored triangulations. Classic results and techniques of crystallizations are reviewed from this point of view. A new set of combinatorial invariants of manifolds is defined, and related to the fundamental group and other known invariants. A universal group theoretic approach for this theory is introduced.

Decomposability of polytopes and polyhedra

Abstract: In this paper decomposability of polytopes (and polyhedral sets) is studied by investigating the space of affine dependences of the vertices of the dual polytope. This turns out to be a fruitful approach and leads to several new results, as well as to simpler proofs and generalizations of known results. One of the new results is that a 3-polytope with more vertices than facets is decomposable; this leads to a characterization of the decomposability of 3-polytopes.

Degenerate submanifolds in Semi-Riemannian geometry

Abstract: Degenerate submanifolds of Semi-Riemannian manifolds are studied. The relation of the Semi-Riemannian structure of a Semi-Riemannian manifold M to the intrinsic singular Semi-Riemannian structure of a degenerate submanifold H in M is investigated. Gauss-Codazzi equations are obtained for a certain class of degenerate submanifolds of Semi-Riemannian manifolds.

Non-classical triangle buildings

Abstract: In [7], J. Tits classifies the affine buildings of rank greater or equal to 4. That leaves the question of the rank 3 buildings. This paper shows that the class of all triangle buildings is as wild as the class of all projective planes.

2-Type surfaces in S 3

Abstract: It is proved that the Riemannian product of two plane circles of different radii are the only compact surfaces of the 3-spheres in R4 which can be constructed in R4 by using eigenfunctions associated with two different eigenvalues of their corresponding Laplacians.

A structure theorem for weak buildings of spherical type

Abstract: Following earlier work of Tits [8], this paper deals with the structure of buildings which are not necessarily thick; that is, possessing panels (faces of codimension 1) which are contained in two chambers, only. To every building Δ, there is canonically associated a thick building \(\bar \Delta \) whose Weyl group W(\(\bar \Delta \)) can be considered as a reflection subgroup of the Weyl group W(Δ) of Δ. One can reconstruct Δ from \(\bar \Delta \) together with the embedding W(\(\bar \Delta \)) ↪ W(Δ). Conversely, if \(\bar \Delta \) is any thick building and W any reflection group containing W(\(\bar \Delta \)) as a reflection subgroup, there exists a weak building Δ with Weyl group W and associated thick building \(\bar \Delta \).

Regular genus: The boundary case

Abstract: A particular kind of 2-cell imbedding for a class of edge-coloured graphs into surfaces with boundary is introduced and studied. This allows to define, as in [13], where the closed case was traited, a pair of invariants — the regular genus and the hole-number — for every n-manifold with boundary. These invariants are proved to coincide with the classical ones in dimension two, and to be strictly related with a Heegaard-like handlebody decomposition in dimension three. A characterization of 261-1 concludes the work.

Cyclic polytopes and d -order curves

Abstract: A cyclic d-polytope is a convex polytope combinatorially equivalent to the convex hull of a finite subset of a d-order curve in Rd. We give an affirmative answer to a conjecture of M. A. Perles [4] by proving that every even-dimensional cyclic polytope occurs in this way: its set of vertices can always be extended to a d-order curve.

Linear independence in finite spaces

Abstract: The maximum number m2(n, q) of points in PG(n, q), n⩾2, such that no three are collinear is known precisely for (n, q)=(n,2), (2,q), (3,q), (4, 3), (5,3). In this paper an improved upper bound of order q n−1 −1/2qn−2 is obtained for q even when n⩾4 and q>2. A necessary preliminary is an improved upper bound for m′2(3, q), the maximum size of a k-cap not contained in an ovoid. It is shown that \(m'_2 (3,q){\text{ }} \leqslant q^2 - \tfrac{1}{2}{\text{q}} - \tfrac{1}{2}\sqrt {\text{q}} {\text{ + 2}}\) and that m′2(3, 4)=14.

A characterization of exterior lines of certain sets of points in PG (2, q )

Abstract: Let A and B be disjoint sets of points in PG(2, q) the Desarguesian projective plane of order q, with |A|≥q, |B|=q+1, such that each line through a point of A meets B (just once). Then B is a line.

On a projectively minimal hypersurface in the unimodular affine space

Abstract: We derive a differential equation defining a projectively minimal hypersurface in the affine space A n+1. Several examples of such hypersurfaces are given. We prove that only an ellipsoid is a projectively minimal surface in A 3 which is compact and strongly convex.

An inequality linking packing and covering densities of plane convex bodies

Abstract: For every convex body K in R2, let δ(K) denote the packing density of K, i.e. the density of the tightest packing of congruent copies of K in R2, and let υ(K) denote the covering density of K, i.e. the density of the thinnest covering of R2 with congruent copies of K. It is shown here that 4δ(K)⩾3υ(K) for every convex body K in R2. This inequality is the strongest possible, since if E is an ellipse, then the equality 4δ(E)=3υ(E) holds. Two corollaries are presented, and a summary of known bounds for packing and covering densities is given.

On rosettes and almost rosettes

Abstract: The closed plane curves of class C 2 which have curvature k(s) > 0 or k(s) ≥ 0 with a finite number of zeros are studied The results concern the existence of normal lines which divide the perimeter into equal parts and the existence of some special kinds of pairs of points on these curves as orthodiameter pairs, antipodal pairs, etc The paper also contains some generalizations of the theorems of Blaschke-Suss and Barbier

SU4 (ℂ) als Kollineationsgruppe in sechzehndimensionalen lokalkompakten Translationsebenen

Abstract: We study 16-dimensional locally compact translation planes in which, for an affine point o, the stabilizer $$\mathbb{G}_o $$ of the affine collineation group $$\mathbb{G}$$ contains a subgroup Σ locally isomorphic to SU4 (ℂ). If ⌆ has only one affine fixed point o, then it is shown that either the plane is the classical Moufang plane over the Cayley numbers, or else Σ must be normal in the stabilizer $$\mathbb{G}_o $$ and $$\mathbb{G}$$ has dimension at most 37. This also comprises the proof of the fact that if $$\mathbb{G}_o $$ contains a subgroup locally isomorphic to SU4(ℂ) × SL2(ℂ) then the plane is the classical Cayley plane. The case that ⌆ has more affine fixed points in dealt with as well; then, except for a well-known family of planes admitting Spin7(ℝ) as a group of collineations, $$\mathbb{G}$$ has dimension at most 34.

A class of non-algebraic matroids of rank three

Abstract: We obtain an infinite class of simple non-algebraic matroids of rank 3, the minors of which are vector matroids and therefore algebraic We prove that the matroids are non-algebraic with the aid of a theory of harmonic conjugates in full algebraic combinatorial geometries [7]

A new result on difference sets with -1 as multiplier

Abstract: In this paper we study finite abelian groups admitting a difference set with multiplier -1. In these groups we have that each integer, which is relatively prime to the group order, is a multiplier (see [1] and Section 1 of this paper).

Quadrilaterals and pentagons in arrangements of lines

Abstract: Let pk denote the number of k-sided faces in an arrangement of n≥5 lines in the real projective plane. B. Grunbaum has shown that p4≤1/2n(n−3) and has conjectured that equality can occur only for simple arrangements. We prove this conjecture here. We also show that 4p4+5p5≥3n holds for every simple arrangement of n≥4 lines. This latter result is a strengthening of a theorem of T. O. Strommer.

Geometrie du plan projectif des octaves de Cayley

Abstract: In this article we study certain geometric aspects of the projective plane P2(O) over the octaves (Cayley numbers) over the reals. First, we use the explicit representation of points of P2(O) by Hermitian 3×3 matrices over the octaves to determine homogeneous coordinates on the projective line with the help of the fibration of S15 with basis S8 and fiber S7. Next we give a table of Lie products in the Lie algebra F4, which enables us to explicitly compute the curvature tensor of P2(O) as a symmetric space. Finally we exhibit a non-zero skew symmetric 8-form which is invariant under the holonomy group Spin(9). The expression we obtain is the analog of the Kahler form and the fundamental 4-form on the complex and quaternion projective plane, respectively.

A characterization theorem for certain unions of two starshaped sets in R 2

Abstract: Let S be a compact set in R2. For S simply connected, S is a union of two starshaped sets if and only if for every F finite, F\( \subseteq \) bdry S, there exist a set G\( \subseteq \) bdry S arbitrarily close to F and points s, t depending on G such that each point of G is clearly visible via S from one of s, t. In the case where ∼S has at most finitely many components, the necessity of the condition still holds while the sufficiency fails.