Horizontal Metric Characterizations of Polar Representations
Let ρ be an orthogonal representation on a Euclidean space V,and SV be the unit sphere of V.Let (d)H and dH be the horizontal metrics on V and SV induced byρ,respectively.Our main result is to show that the following conditions are equivalent:(1)The representation ρ is polar.(2)(V,(d)H)is a CAT(0) space.(3)(SV,dH)is a CAT(1)space.
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