Chow's Theorem for Linear Spaces

TL;DR

This paper extends Chow's Theorem to linear spaces, showing that bijections preserving line intersections are induced by collineations or dualities, with special cases for 3-dimensional generalized projective spaces.

Contribution

It generalizes Chow's Theorem to broader linear spaces, characterizing intersection-preserving bijections as collineations or dualities, especially in 3-dimensional cases.

Findings

Bijections preserving line intersections are induced by collineations or dualities.

The second case occurs only in 3-dimensional generalized projective spaces.

The result extends classical Chow's Theorem to linear spaces with dimension at least 3.

Abstract

If $\phi: L\to L'$ is a bijection from the set of lines of a linear space $(P,L)$ onto the set of lines of a linear space $(P',L')$ ($\dim P, \dim P'\geq 3$), such that intersecting lines go over to intersecting lines in both directions, then $\phi$ is arising from a collineation of $(P,L)$ onto $(P',L')$ or a collineation of $(P,L)$ onto the dual linear space of $(P',L')$. However, the second possibility can only occur when $(P,L)$ and $(P',L')$ are 3-dimensional generalized projective spaces.