Transformations of polar Grassmannians preserving certain intersecting relations

Let $Π$ be a polar space of rank $n\ge 3$. Denote by ${\mathcal G}_{k}(Π)$ 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 ${\mathfrak R}_{i,j}$, $0\le i\le j\le k+1$ formed by all pairs $(X,Y)\in {\mathcal G}_{k}(Π)\times {\mathcal G}_{k}(Π)$ such that $\dim_{p}(X^{\perp}\cap Y)=k-i$ and $\dim_{p} (X\cap Y)=k-j$ ($X^{\perp}$ consists of all points of $Π$ collinear to every point of $X$). We show that every bijective transformation of ${\mathcal G}_{k}(Π)$ preserving ${\mathfrak R}_{1,1}$ is induced by an automorphism of $Π$ and the same holds for the relation ${\mathfrak R}_{0,t}$ if $n\ge 2t\ge 4$ and $k=n-t-1$. In the case when $Π$ is a finite classical polar space, we establish that the valencies of ${\mathfrak R}_{i,j}$ and ${\mathfrak R}_{i',j'}$ are distinct if $(i,j)\ne (i',j')$.

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