Why is GL(n,C)/U(n) a CAT(0) space? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T17:37:20Z http://mathoverflow.net/feeds/question/11816 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11816/why-is-gln-c-un-a-cat0-space Why is GL(n,C)/U(n) a CAT(0) space? Peter Samuelson 2010-01-15T03:20:36Z 2010-01-15T13:34:16Z <p>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.</p> http://mathoverflow.net/questions/11816/why-is-gln-c-un-a-cat0-space/11818#11818 Answer by Andy Putman for Why is GL(n,C)/U(n) a CAT(0) space? Andy Putman 2010-01-15T03:45:00Z 2010-01-15T03:45:00Z <p>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.</p> http://mathoverflow.net/questions/11816/why-is-gln-c-un-a-cat0-space/11862#11862 Answer by Deane Yang for Why is GL(n,C)/U(n) a CAT(0) space? Deane Yang 2010-01-15T13:34:16Z 2010-01-15T13:34:16Z <p>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.</p>