  In this short note, completing a sequence of studies by Cooperstein, Kasikova
and Shult, we consider the k-Grassmannians of a number of polar geometries of
finite rank n. We classify those subspaces that are isomorphic to the
j-Grassmannian of a projective m-space. In almost all cases, these are
parabolic, that is, they are the residues of a flag of the polar geometry.
Exceptions only occur when the subspace is isomorphic to the Grassmannian of
2-spaces in a projective m-space and we describe these in some detail. This
Witt-type result implies that automorphisms of the Grassmannian are almost
always induced by automorphisms of the underlying polar space.
