Another response to Ravenel question and then a followup question....DMD _________________________________________________ Date: Thu, 28 Sep 2000 12:52:14 +0200 From: Julius Korbas Subject: Re: question aby Grassmanians A possible reply to the question of D. Ravenel (but perhaps it is not that what he needs): Take the Grassmannian of k-dimensional vector subspaces in Euclidean n-space a s the manifold of real symmetric idempotent n x n- matrices with trace k (this interpretation can be generalized, see e.g. V. Bartik, J. Korbas, Arch. Math. (Basel) 60 (1993), 563-567). Best regards, Julius Korbas (korbas@fmph.uniba.sk) Department of Algebra, Faculty of Mathematics and Physics, Comenius University, Mlynska dolina, SK-842 15 Bratislava, Slovakia _____________________________________________________________ Date: Thu, 28 Sep 2000 11:48:33 -0400 (EDT) From: "Douglas C. Ravenel" Subject: Followup on Grassmannians Thanks to Tom Goodwillie et al for responses to my previous question. Here are two followup questions. 1. Let G(m,n) denote the Grassmannian of m-dimensional subsapces of R^n. In terms of the metric defined below, what is its volume? 2. I can think of another way to define a metric on G(m,n). Each subspace gives us a compact subset of the unit ball in R^n. One has the Hausdorf metric on the set of all nonempty compact subsets of R^n defined as follows. The distance from A to B to be the samllest number r such that each point in A is within r of some point in B and vice versa. Is this metric on G(m, n) a scalar multiple of the one below? Doug Ravenel On Thu, 28 Sep 2000, DON DAVIS wrote: > Subject: Re: question aby Grassmanians > Date: Wed, 27 Sep 2000 22:25:58 -0400 (EDT) > From: "Tom Goodwillie,304 Kassar,863-2590,617-926-3565" > > How about identifying a vector subspace of R^n with the orthogonal > projection onto it? Thus the Grassmanian becomes a submanifold of > the vector space of n by n real matrices, namely the set of all idempotent > symmetric matrices of a given rank. That gives it a metric, which > seems like a pretty nhice one ... > > Tom Goodwillie > > Douglas C. Ravenel, Chair |918 Hylan Building Department of Mathematics |drav@math.rochester.edu University of Rochester |(716) 275-4413 Rochester, New York 14627 |FAX (716) 273-4655 Department of Mathematics home page: http://www.math.rochester.edu/ Personal home page: http://www.math.rochester.edu/u/drav/ Faculty Senate home page: http://www.cc.rochester.edu:80/Faculty/senate/ Math 141 home page: http://www.math.rochester.edu/courses/current/MTH141/