This study is a continuation of my papers, Let M be a compact simply connected riemannian symmetric space of dimension m ( ≥ 2), and s an integer such that 1 ≤ s ≤ m. Let G~s(T_PM) be the set of s-dimensional linear subspaces in a tangent space T_pM and denote by G~S(TM) the Grassmann bundle over M with fibres G~S(T_PM). For an arbitrary subset V in G~S(TM) an s-dimensional connected sub-manifold S of M is called a V-submanifold if at each point p of S the tangent space T_PS belongs to V. The collection of V-submanifolds, denoted by g(M, V), constitutes a V-geometry. The term "Grassmann geometries" in the title is a collected name for such V-geometries and it has been introduced in R. Harvey-H. B. Lawson.
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