A Property of Isometric Mappings Between Dual Polar Spaces of Type {DQ (2n, {\mathbb K})}

Let f be an isometric embedding of the dual polar space ${\Delta = DQ(2n, {\mathbb K})}Let f be an isometric embedding of the dual polar space D = DQ(2n, \mathbb K){\Delta = DQ(2n, {\mathbb K})} into D￠ = DQ(2n, \mathbb K￠){\Delta^\prime = DQ(2n, {\mathbb K}^\prime)}. Let P denote the point-set of Δ and let e￠: D￠? S￠ @ PG(2ⁿ - 1, \mathbb K￠){e^\prime : \Delta^\prime \rightarrow {\Sigma^\prime} \cong {\rm PG}(2^n - 1, {{\mathbb K}^\prime})} denote the spin-embedding of Δ′. We show that for every locally singular hyperplane H of Δ, there exists a unique locally singular hyperplane H′ of Δ′ such that f(H) = f(P) ?H￠{f(H) = f(P) \cap H^\prime}. We use this to show that there exists a subgeometry S @ PG(2ⁿ - 1, \mathbb K){\Sigma \cong {\rm PG}(2^n - 1, {\mathbb K})} of Σ′ such that: (i) e￠°f (x) ? S{e^\prime \circ f (x) \in \Sigma} for every point x of D; (ii) e : = e￠°f{\Delta; ({\rm ii})\,e := e^\prime \circ f} defines a full embedding of Δ into Σ, which is isomorphic to the spin-embedding of Δ.