A Geometry of real Grassmannian manifolds. Part III

A canonical decomposition for an element of the tangent fibration of Grassmannian manifold $G^+_{p,n}$ in its Plücker model is constructed. By means of the decomposition a concept of stationary angles between oriented planes is introduced and a connection with stationary angles in a nonoriented case is ascertained. A direct formula allowed to calculate the diameter and the radius of injectiveness of the manifold $G^+_{p,n}$ is given. A problem of the uniqueness of the above canonical decomposition has been reduced to a previously solved by the author similar problem of the decomposition of bivectors which realizes their mass. By virtue of a developed technique a structure of the closure of an arbitrary geodesic in manifolds $G^+_{p,n}$ and $G_{p,n}$ was determined. The last result for manifolds $G_{p,n}$ was earlier announced by Wong without proof.