Extensions of isomorphisms for affine dual polar spaces and strong parapolar spaces

Let B be a class of point-line geometries. Given Γi ∈ B with subspace Si for i = 1, 2, does any isomorphism Γ1−S1 −→ Γ2−S2 extend to a unique isomorphism Γ1 −→ Γ2? It is known to be true if B is the class of almost all projective spaces or the class of almost all non-degenerate polar spaces. We show that this is true for the class of almost all strong parapolar spaces, including dual polar spaces. A special case occurs when Γ1 = Γ2 = Γ has an embedding into a projective space P(V ) that is natural in the sense that Aut(Γ) ≤ PΓL(V ). Then the question becomes whether P(V ) is also the natural embedding for Γ−S. Our result shows that in most cases the stabilizer StabAut(Γ)(Γ−S) is faithful on Γ−S and equals Aut(Γ−S) and so the answer is affirmative. We know that there exist some interesting exceptions. These will be covered in a subsequent paper.