Showing papers in "Geometriae Dedicata in 1985"

Structure theorems on riemannian spaces satisfying R(X, Y) · R =0,

Abstract: Introduction The curvature tensor R of a locally symmetric Riemannian space satisfies R(X, Y) R — 0 for all tangent vectors X and 7, where the linear endomorphism R(X, Y) acts on R as a derivation. This identity holds in a space of recurrent curvature also. The spaces with R(X9 Y) R = 0 have been investigated first by E. Cartan [2] as these spaces can be considered as a direct generalization of the notion of symmetric spaces. Further on remarkable results were obtained by the authors A. Lichnerowicz [13], R. S. Couty [3], [4] and N. S. Sinjukov [19], [20], [21]. In one of his papers K. Nomizu [15] conjectered that an irreducible, complete Riemannian space with dim > 3 and with the above symmetric property of the curvature tensor is always a locally symmetric space. But this conjecture was refuted by H. Takagi [22] who constructed 3-dimensional complete irreducible nonlocally-symmetric hypersurfaces with R(X, Y) R — 0. These two papers were very stimulating for the further investigations. We also have to mention the following authors in this field: S. Tanno [23], [24], [25], K. Sekigawa [16], [17] and P. I. Kovaljev [9], [10], [11]. In the following we call a space satisfying R(X, Y) R = 0 a semi-symmetric space. The main purpose of this paper is to determine all semi-symmetric spaces in a structure theorem. In §1 we give local decomposition theorems using the infinitesimal holonomy group, and in §2 we give some basic formulas. We would like to make it perfectly clear that the results of these chapters are concerning general Riemannian spaces, and not only semi-symmetric spaces. In §3 we construct several nonsymmetric semi-symmetric spaces and in §4 we show that every semi-symmetric space can be decomposed locally on an everywhere dense open subset into the direct product of locally symmetric spaces and of the spaces constructed in §3.

Small complete arcs in galois planes

Abstract: In this paper we construct a class of k-arcs in PG(2, q), q=p h, h>1, p≠3 and prove its completeness for h large enough. The main result states that this class contains complete k-arcs with $$k \leqslant 2 \cdot q^{{9 \mathord{\left/ {\vphantom {9 {10}}} \right. \kern- ulldelimiterspace} {10}}} {\text{ }}\left( {10{\text{ divides }}h{\text{ and }}q{\text{ }} \geqslant {\text{ }}q_{\text{0}} } \right).$$ Such complete k-arcs are the unique known complete k-arcs with $$k \leqslant {q \mathord{\left/ {\vphantom {q 4}} \right. \kern- ulldelimiterspace} 4}.$$

The completion problem for partial packings

Abstract: Packings (resolutions) of designs have been of interest to combinatorialists in recent years as a way of creating new designs from old ones. Line packings of projective 3-space were the first packings studied, but it is still unknown when a partial packing can be completed to a packing in this case. In this paper we show that there is no guarantee from a combinatorial point of view of completing such a partial packing even when the deficiency is 2. In particular, we construct for every odd prime power q a set of 2(q 2+1) lines which doubly cover the points of PG(3,q) and yet cannot be partitioned into two spreads (resolution classes). The method is based on manipulations of primitive elements of finite fields.

The weakly neighborly polyhedral maps on the torus

Abstract: A weakly neighborly polyhedral map (w.n.p. map) is a 2-dimensional cell-complex which decomposes a closed 2-manifold without a boundary, such that for every two vertices there is a 2-cell containing them. We prove that there are just five distinct w.n.p. maps on the torus, and that only three of them are geometrically realizable as polyhedra with convex faces.

The uniqueness of Hering's translation plane of order 27

Abstract: It is shown that the following conjecture of Kallaher and Ostrom [2] is correct: Hering's translation plane of order 27 is the only translation plane of odd dimension over its kernel which has a collineation group isomorphic to SL(2, w) with w prime to 5 and to the characteristic, and having no affine perspectivity.

On a generalization of Kantor's likeable planes

Abstract: Generalizing an idea of Kantor [7], Johnson and Wilke [5] introduced ‘elusive’ sets of functions over GF(q) to represent translation planes of order q2 that admit a collineation group of order q2 in the linear translation complement and whose kern contains GF(q). In this paper we determine explicitly all elusive sets for q even. We obtain another translation plane of order 82.