Geometriae Dedicata

, Volume 86, Issue 1, pp 81-91

First online:

On Jordan Angles and the Triangle Inequality in Grassmann Manifolds

  • Yurii A. NeretinAffiliated withInstitute of Theoretical and Experimental PhysicsIndependent University of Moscow

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Let L, M and N be p-dimensional subspaces in \(\mathbb{R}\) n . Let {ψ j } be the angles between L and M, let {ψ j } be the angles between M and N, and let {θ j } be the angles between L and M. Consider the orbit of the vector ψ = (ψ1,...., ψ n ) ∈ \(\mathbb{R}\) p with respect to permutations of coordinates and inversions of axes. Let Z be the convex hull of this orbit. Then θ ∈ ϕ + Z. We discuss similar theorems for other symmetric spaces. We also obtain formula for geodesic distance for arbitrary invariant convex Finsler metrics on classical symmetric spaces.

symmetric spaces matrix inequalities compound distance Finsler metrics