Topological properties of ordinary nematics in 3-space

This paper describes the topologically possible global defect behavior of ordinary nematics in 3-space. It is written for physicists interested in defects of ordered media as well as for topologists, but instead of using an intermediate way of presentation, which might appeal to no one, we first state the result for physicists and then, discussing the proof, turn to mathematicians and physicists who are inclined to read a mathematical paper.     

Partial parallelisms in finite projective spaces

We show the existence of a parallelism of P–U, where P is a finite projective space and U is a subspace of P with dim P–dim U=2 ⁱ . As a consequence we prove a lower bound for the maximum number of disjoint spreads of P.To Helmut Salzmann on the occasion of his 60th birthday     

Subcanonical curves and complete intersections in projective 3-space

Summary Un classico teorema di G. Gherardelli afferma che una curvaC P ³ è intersezionè completa se e soltanto se è proiettivamente normale e sottocanonica. Qui si prova che, seC e a- sottocanonica ed inoltre le superficie di grado 1 + (a/2) (a pari) ovvero (a + l)/2 o (a + 3)/2 o (a + 5)/2 (a dispari) tagliano suC serie complete, alloraC è intersezione completa. Si determina inoltre un bound d funzione di a tale che, seC è a-sottocanonica e di grado d d, alloraC è intersezione completa se e soltanto se le superficie di grado a tagliano suC una serie completa. Si discutono poi numerosi esempi di curve sottocanoniche non intersezioni complete.Paper written while P. Valabrega was member of C.N.R. (G.N.A.S.G.A.) and both authors were supported by M.P.I. funds.     

Bidegree (2,1) parametrizable surfaces in projective 3-space

The main purpose of this paper is to study projective classes of bidegree (2,1) parametrizable surfaces in a real projective 3-space. It turns out that the implicit degree of such surfaces is two, three, or four, and singular curves have degree three. We describe all possibilities for singular curves and pinch points on such surfaces. The presentation has been made from the point of view of analytic geometry and does not presume a deep knowledge of algebraic geometry. Partially supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 38, No. 3, pp. 379–402, July–September, 1998.     

Topological projective geometries

A concept of topological projective geometry is defined, which in contrast to the definitions given in [Mi] and [SÖ] does not contain any dimensional restrictions. Besides elementary properties it is shown in this paper that these topological geometries always possess a coordinatization over a uniquely determined topological division ring if the dimension is finite.Dedicated to Prof. Dr. Hanfried Lenz on his 70^th birthday     

Existence of parallelisms and projective extensions for strongly n-uniform near affine Hjelmslev planes

Geometriae Dedicata - An affine Hjelmslev plane is a near affine Hjelmslev plane with a parallelism. It is proved that every strongly n-uniform near affine Hjelmslev plane possesses an even...     

Finding ein components in the moduli spaces of stable rank 2 bundles on the projective 3-space

Some method is proposed for finding Ein components in moduli spaces of stable rank 2 vector bundles with first Chern class c₁ = 0 on the projective 3-space. We formulate and illustrate a conjecture on the growth rate of the number of Ein components in dependence on the numbers of the second Chern class. We present a method for calculating the spectra of the above bundles, a recurrent formula, and an explicit formula for computing the number of the spectra of these bundles.     

Gluing equations for real projective structures on 3-manifolds

Geometriae Dedicata - Given an orientable ideally triangulated 3-manifold M, we define a system of real valued equations and inequalities whose solutions can be used to construct projective...     

Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions

In this paper, we use Chas–Sullivan theory on loop homology and Leray–Serre spectral sequence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the resonance identity of non-contractible homologically visible prime closed geodesics on such spaces provided the total number of distinct prime closed geodesics is finite.     

The structure of projective maps between real projective manifolds

In this paper we study the set of projective maps between compact properly convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When domain is irreducible and the target is strictly convex, our results imply that each non-trivial homotopy class contains at most one projective map. These results are motivated by the theory of holomorphic maps between compact complex manifolds.     

Topological projective planes and uniform valuations

The valuation topology of any uniformly valued ternary field (K, T, v) can be extended to the projective plane II over (K, T) making it a topological projective plane in the sense of Salzmann. Appealing to Prieß-Crampe's celebrated fixed point theorem for ultrametric spaces, our result allows us to present a wide variety of new, totally disconnected, compact and non-compact topological projective planes.Dedicated to Professor S. Prieß-Crampe on the occasion of her 60th birthday     

On the genus of real projective spaces

We construct a crystallization of the real projective space whose associated contracted complex is minimal with respect to the number of n-simplexes. Then we compute the regular genus of , which is the minimum genus of a closed connected surface into which a crystallization of regularly embeds. Received: 7 February 2007     

Deforming convex real projective structures

In this paper we define new flows—eruption flows and internal bulging flows—on the deformation space of convex real projective structures. These flows are associated to internal parameters associated to a pair of pants decomposition.     

A constructive real projective plane

The classical theory of plane projective geometry is examined constructively, using both synthetic and analytic methods. The topics include Desargues’s Theorem, harmonic conjugates, projectivities, involutions, conics, Pascal’s Theorem, poles and polars. The axioms used for the synthetic treatment are constructive versions of the traditional axioms. The analytic construction is used to verify the consistency of the axiom system; it is based on the usual model in three-dimensional Euclidean space, using only constructive properties of the real numbers. The methods of strict constructivism, following principles put forward by Errett Bishop, reveal the hidden constructive content of a portion of classical geometry. A number of open problems remain for future studies.