Two questions in the exterior geometry of Plucker embeddings for Grassmannian manifolds

For the Plucker model of the Grassmannian manifold $G^+_{p,p+q}$ a structure of the intersection of the manifold with its own tangent space at an arbitrary point considered as a subspace of the exterior algebra is described. A direct formula for the second main form of the manifold $G^+_{2,4}$ as a hypersurface in a five-dimensional sphere is given. Sets of the constancy for the function of the normal curvature for this hypersurface are studied.