Geometry of real Grassmanian manifolds. IV

A classical geodesic immersion of the Grassmanian manifold $G_p^+(V)\subset\Lambda(V)$ is described by means of the exterior algebra $\Lambda(V)$. The group $I_0(G^+)$ of isometries of the Grassmanian is described without the theory of Lie groups and algebras. Some interior and exterior properties of Grassmanian manifolds are proved by means of the invariant Plücker immersion. The isomorphic type of the group of rotations of the Grassmanian manifold around one of its geodesic lines is also described.