3 more on Grassmanians, from Korbas, Meyer, and Ravenel......DMD ____________________________________________________ Date: Mon, 02 Oct 2000 09:26:56 +0200 From: Julius Korbas Subject: Re: Grassmanians revisited Having looked a little at the discussion around D. Ravenel's question, I think the following book can be of some use to those who can read in Russian (it has a section on metric properties of real Grassmann manifolds): Ivanov, L.D.: Variations of sets and functions (in Russian), "Nauka", Moscow 1975. Best regards, Julius Korbas (korbas@fmph.uniba.sk) KATC MFF UK, Mlynska dolina, SK-842 48 Bratislava, Slovakia ___________________________________________________ Date: October 2, 2000 From: Jean-Pierre Meyer Subject: Re: 2 more on Grassmanians Concerning Doug's question on Grassmanians, there is a survey article entitled "Grassmann manifolds and the Grassmanian image of submanifolds" by Borisenko and Nikolaevski in Uspekhi Mat. Nauk 462 (1991), 41-83 or Russian Math Surveys 46:2 (1991) 45-94. While the bulk of this paper deals with differential-geometric matters irrelevant to the query, the introductory portions discuss metrics and Riemannian metrics on G(m,n). Jean-Pierre Meyer ______________________________________________ Date: Mon, 2 Oct 2000 18:57:08 -0400 (EDT) From: "Douglas C. Ravenel" Subject: Third question on Grassmanians Thanks again to all of you who responded to my earlier queries about metrics on Grassmannians. Here is a followup to my earlier question about volumes. Naively it seems that if one has a fibre bundle $$ F ----> E ----> B $$ of Riemannian manifolds, a metric can be chosen on the total space $E$ which is locally the product of th metrics on the fibre $F$ and base $B$. Hence if all manifolds are compact, then the volume of $E$ is the product of the volumens of $B$ and $F$. Moreover if $F$ is a compact Lie group acting freely and isometrically on $E$, then a metric on $B$ can be chosen so that a similar volume formula holds. IS THIS TRUE? For example, applying this method to the Hopf fibration $$ S^1 ---> S^3 ---> CP^1 $$ with the standard metrics on $S^1$ and $S^3$ leads to the conclusion that the induced metric on $CP^1$ is isometric to a 2-sphere of radius 1/2. Assuming this method is valid, one could study the fibrations $$ SO(n-1) ----> SO(n) ----> S^{n-1} $$ by induction on $n$ and arrive at volumes for $SO(n)$ (the sepcial orthogonal groups) from those of the the spheres. (A formula for the volume of $S^{n-1}$ can be found in Coxeter's book on regular polytopes.) For example we could deduce that $SO(3)$ has volume $8\pi^2$, so $Spin(3)$ is isomteric to a 3-sphere of radius 2. Then one could use the fact that $$ G(m,n) = SO(n)/SO(m)\times SO(n-m) $$ to find the volume of the Grassmannian $G(m,n)$ with respect to a suitable metric described abstractly above. The answer would be a certain rational multiple of a certain power of $\pi$. For example, $G(2,4)$ would have volume $4\pi^2$, which is $3/2$ the volume of $S^4$, but the volume of $CP^2$ would be just $\pi^2/2$. 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/