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A Characteristic Property of Elliptic Plücker Transformations

Hans Havlicek

Abstract

We 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".

A Characteristic Property of Elliptic Plücker Transformations

Abstract

We 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".

Paper Structure

This paper contains 3 sections, 14 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction and Main Results
  2. Clifford Parallelism
  3. Proofs

Key Result

THEOREM 1

Let $({\cal P},{\cal L},\pi)$ be a $3$-dimensional classical elliptic space. If $\hbox{$\varphi\,:\,{\cal L}\rightarrow{\cal L}$}$ is a bijection satisfying then $\varphi$ is an elliptic Plücker transformation.

Figures (1)

  • Figure :

Theorems & Definitions (14)

  • THEOREM 1
  • THEOREM 2
  • PROPOSITION 1
  • PROPOSITION 2
  • PROPOSITION 3
  • PROPOSITION 4
  • PROPOSITION 5
  • PROPOSITION 6
  • PROPOSITION 7
  • PROPOSITION 8
  • ...and 4 more