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Orthogonal invariant Riemannian metrics on real Grassmann manifolds. (Russian. English summary) Zbl 0967.53034
Author’s abstract: A full description of the 2-parameter family of all possible \(SO(4)\)-invariant Riemannian metrics on the real Grassmann manifolds \(G_{2,4}\) and \(G^+_{2,4}\) 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 \((p,n)\neq (2,4)\) 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 \({n\choose p}\)-space), which allows us to solve the problems of intrinsic geometry of Grassmann manifolds by methods of exterior geometry.

53C30 Differential geometry of homogeneous manifolds