Orthogonal invariant Riemannian metrics on the real Grassmann manifolds

A full description of the $2$-parameter family of all possible $SO(4)$-invariant Riemannian metrics on the real Grassmann manifolds $G_{2,4}$ и $G_{2,4}^+$ and is given and an extremal property characterizing the canonical metric on $G_{2,4}$ is described. On the basis of these results, we give a new short geometrical proof of the uniqueness (up to the constant factor) of invariant metrics on $G_{p,n}$ and $G_{p,n}^+$ for $(p,n)\ne(2,4)$ and construct these metrics. We use the embeddings of the Grassmann manifolds in the polivector space $\Lambda_{p,n}$ (which can be identified as the Euclidean $\bigl(\frac{n}{p}\bigr)$-space), which allows us to solve the problems of intrinsic geometry of Grassmann manifolds by methods of exterior geometry.