Classical polar spaces (sub–)weakly embedded in projective spaces
نویسنده
چکیده
In this paper we are concerned with classical polar spaces, i. e. with the set of points and lines of some vector space W on which a non–degenerate (σ, )-hermitian form or pseudo-quadratic form vanishes. To state the Main Theorem we introduce some notation. Let L be a division ring and W be a (left–)vector space over L endowed with a (σ, )–hermitian form or a pseudo–quadratic form q (with associated (σ, )–hermitian form f) in the sense of [Ti, §8]. We may assume that = ±1 and σ = id. We let
منابع مشابه
The geometry of secants in embedded polar spaces
Consider a polar space S weakly embedded in a projective space P. A secant of S is the intersection of the point set of S with a line of P spanned by two non-collinear points of S. The geometry consisting of the points of S and as lines the secants is a so-called Delta space. In this paper we give a characterization of this and some related geometries.
متن کاملPolar spaces embedded in projective spaces
Let P be a polar space induced by a quasipolarity on a projective space G. In this article, we give sufficient conditions for an embedding (i.e. an incidence-preserving map which is injective on points and lines) of P into a projective space H to be extendable to an incidencepreserving map from G to H. As a consequence, we obtain conditions for such an embedding to be induced by a semilinear ma...
متن کاملOn Totally Umbilical Hypersurfaces of Weakly Projective Symmetric Spaces
The object of the present paper is to study the totally umbilical hypersurfaces of weakly projective symmetric spaces and it is shown that the totally umbilical hypersurfaces of weakly projective symmetric space is also a weakly projective symmetric space.
متن کاملPartial ovoids and partial spreads of classical finite polar spaces
We survey the main results on ovoids and spreads, large maximal partial ovoids and large maximal partial spreads, and on small maximal partial ovoids and small maximal partial spreads in classical finite polar spaces. We also discuss the main results on the spectrum problem on maximal partial ovoids and maximal partial spreads in classical finite polar spaces. 1 Classical finite polar spaces Th...
متن کاملOn the incidence structures of polar spaces and quadrics
In 1969 Buekenhout characterized non-singular quadrics of finite-dimensional projective spaces as polar spaces spanning the whole space and containing every line of the projective space which they intersect in at least three points (see [2]). Using suitable synthetic properties of the pairs of non-collinear disjoint lines, in this paper I present a new characterization of polar spaces and a com...
متن کامل