Characterization of embedded special manifolds

Abstract

A semilinear space is a pair ( G, F) , where G is a set, whose elements are called points, and F is a collection of subsets of G . The elements of F are lines. The axioms defining a semilinear space require that each line contains at least two points, each point is contained in at least one line and for every two distinct points there is at most one line which contains both points. In this paper we regard the classical manifolds as structures formed only by points and lines, i.e. as semilinear spaces. Geometric characterizations of such spaces have been given by several authors, and some of them put special emphasis on the topological and order structures. We combine these results and the ones concerning the embeddings of the Grassmann spaces and product spaces and obtain intrinsic properties which characterize, up to projections, the embedded Grassmann and Segre manifolds. The results of this work are offered as a solution to the following question studied by Giuseppe Tallini: Describe the classical manifolds, which are ‘rich’ structures, by essential and elementary geometric properties.