From: lrudolph@panix.com (Lee Rudolph)
Newsgroups: sci.math
Subject: Favorite Quaternion Facts #17
Date: 1 Mar 1995 08:39:48 -0500
An easy way to see that the Grassmannian G of oriented 2-dimensional
vector subspaces V of R^4 is the Cartesian product of two copies of
the 2-dimensional sphere S^2 is to do a few simple calculations
which verify that, given such a V, there is one and only one pair
(p,q) of unit pure quaternions such that V is a complex line
both in the complex structure on R^4 (identified in the usual
way with the quaternions) in which J (i.e., "multiplication
by i"--the complex number, NOT the quaternion) is quaternionic
left multiplication by p, and in the complex structure in
which J is quaternionic right multiplication by q.
(The 2-sphere of unit pure quaternions, i.e., quaternions
of norm 1 with real part 0, is the set of square roots of
-1 in the quaternions. Notice that, if you orient R^4 first,
then one of those two complex structures is inconsistent with
that orientation; so don't bother.)
Lee Rudolph