Geometry of self-dual flats over a PID on a polarity

Abstract

Abstract Let R be any commutative principal ideal domain (PID), and let n be an integer ≥ 2. Denote by 𝕀 n − 1 the set of all (n − 1)-dimensional self-dual flats of a polarity in the projective geometry ℙ(R 2n ). The geometric character of 𝕀 n − 1 is discussed. Two self-dual flats A, B ∈ 𝕀 n − 1 are said to be adjacent if dim(A ∩ B) = n − 2. We prove that ϕ : 𝕀 n − 1 → 𝕀 n − 1 is an adjacency preserving surjection if and only if ϕ is an adjacency preserving bijection in both directions. Chow's theorem on the self-dual flats is extended as follows: If the polarity is the symplectic polarity and ϕ : 𝕀 n − 1 → 𝕀 n − 1 is an adjacency preserving surjection, then ϕ is induced by a collineation on ℙ(R 2n ).