A characteristic property of elliptic Pl�cker transformations

Abstract

Dedicated to Walter Benz on the occasion of his 65th birthdayWe discuss elliptic Plücker transformations of three-dimensional elliptic spaces. These are permutations on the set of lines such that any two related (orthogonally intersecting or identical) lines go over to related lines in both directions. It will be shown that for "classical" elliptic 3-spaces a bijection of its lines is already a Plücker transformation, if related lines go over to related lines. Moreover, if the ground field admits only surjective monomorphisms, then "bijection" can be replaced by "injection".

Orthogonality of subspaces in metric-projective geometry

“…In [8] Havlicek proves that ortho-adjacencypreserving transformations of elliptic spaces with dimensions other than 3 are induced by orthogonality-preserving collineations. Later, in [10], he completes his result for 3-dimensional spaces. A similar result for symplectic spaces is also due to Havlicek in [9] and for hyperbolic spaces is due to List in [14].…”

Coplanarity of lines in projective and polar Grassmann spaces

“…In metric geometry lines intersecting at right angles play an essential role. It has been proved in [8,11] for elliptic spaces, in [12] for symplectic spaces, and in [19][20][21][22]24] for hyperbolic spaces that transformations which preserve ortho-adjacency of lines are induced by collineations that preserve orthogonality, unless the underlying projective space has three dimensions. These results were generalized for k-subspaces in metric-projective settings in [28].…”