You might find it interesting to think of CP^2 as a "qutrit": rank one 3x3 Hermitian matrices with unit trace. At least now you have another search term! Example: http://arxiv.org/abs/1111.4427. Physicists often relax the rank one property, in which case the qutrit is in a "mixed-state" (as opposed to a "pure state"). -Veit On Aug 28, 2013, at 12:02 AM, Dan Asimov <dasimov@earthlink.net> wrote:
As you may know, the complex projective plane (CP^2) is defined as the equivalence classes of C^3 - {(0,0,0)} under the equivalence relation (x,y,z) ~ (cx,cy,cz) for any nonzero complex number c.
Which is the same as letting S^5 = {(x,y,z) in C^3 | |x|^2 + |y|^2 + |z|^2 = 1}, factored out by the equivalence relation (x,y,z) = (ux,uy,uz), where u is any complex number with |u| = 1. Each equivalence class is clearly a great circle of S^5.
So, CP^2 is a 4-manifold. (Some people like to think of it as a complex 2-manifold, but be that as it may.)
The standard metric on CP^2 is defined from this last version, with the distance between points of CP^2 being defined by the distance between the corresponding great circles of S^5. (Every point of either circle has the same distance to the other circle -- through S^5 -- as any other point of it does.)
It's easy to see that this geometry must be homogeneous: for any 2 points p, q of CP^2, there is an isometry of CP^2 taking p to q. (It's not isotropic, though.)
With respect to this standard metric, it's interesting to ask what happens as you get farther and farther from a given point p of CP^2. Turns out that the locus of all points of CP^2 at a distance D from p form a 3-sphere S^3, for 0 < D < pi/2. As D increases towards pi/2, the size of this S^3 increases toward a supremum of radius 1.
But the supremum is never reached. Because suddenly at D = pi/2, the locus changes from a 3-sphere to a 2-sphere (of radius = 1/2). And pi/2 is the maximum distance in CP^2 (i.e., its diameter), so you can't get any farther from p. And this must be true no matter which p we started with.
This seems to be almost paradoxical behavior for a manifold that is homogeneous.
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