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The title says it all. In one of his answers to the question "Convex hull in CAT(0)" (I don't have the points to post a link, if someone doesn't mind link-ifying this that would be cool), Greg Kuperberg said that GL(n,C)/U(n) is a CAT(0) space. I was wondering why this is true, or if there's a reference for this.

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up vote 2 down vote accepted

It also suffices to check that the sectional curvature of this space, using the Riemannian metric induced by the Killing form, is nonpositive. I recommend that you both figure out how to do the explicit calculation of the sectional curvature for this particular example and learn the general theory referred to in Andy's answer. They are, of course, essentially the same answer.

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This space is a symmetric space of noncompact type, which are all CAT(0). See the chapter on symmetric spaces in Bridson-Haefliger. I think it is also contained in the first chapter of Eberlein's book on manifolds of nompositive curvature.

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For the similar space GL(n,R)/SO(n), a detailed and accessible discussion is given in Ch.12 of Lang's Fundamentals of Differential Geometry. – Jairo Bochi Sep 4 '11 at 12:23

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