Transformations of polar Grassmannians preserving certain intersecting relations

Let Π be a polar space of rank n≥3. Denote by \({\mathcal{G}}_{k}(\varPi)\) the polar Grassmannian formed by singular subspaces of Π whose projective dimension is equal to k. Suppose that k is an integer not greater than n?2 and consider the relation \({\mathfrak{R}}_{i,j}\) , 0≤i≤j≤k+1, formed by all pairs \((X,Y)\in{\mathcal{G}}_{k}(\varPi)\times{\mathcal{G}}_{k}(\varPi)\) such that dim _p (X ^⊥∩Y)=k?i and dim _p (X∩Y)=k?j (X ^⊥ consists of all points of Π collinear to every point of X). We show that every bijective transformation of \({\mathcal{G}}_{k}(\varPi)\) preserving \({\mathfrak{R}}_{1,1}\) is induced by an automorphism of Π, except the case where Π is a polar space of type D _n with lines containing precisely three points. If k=n?t?1, where t is an integer satisfying n≥2t≥4, we show that every bijective transformation of \({\mathcal{G}}_{k}(\varPi)\) preserving \({\mathfrak{R}}_{0,t}\) is induced by an automorphism of Π.