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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.

MSC:
 53C30 Differential geometry of homogeneous manifolds