Fubini-Study metric, revisited

There was an old thread here in 2000 about the Fubini-Study metric and
what it *means* at

news:Fo5J23.MGK@xxxxxxxxxxxxx
http://groups.google.com/group/sci.physics.research/browse_frm/thread/128a4d44891cc874

In that thread, John Baez wrote

> There's a nice way to [figure out a good way to measure the distance
> between lines through the origin in Euclidean space] starting from the
> usual way to measure the distance between points in Euclidean space.
> If you have the knack for math, you may be able to figure out the
> trick for yourself. Then the generalization to complex projective
> space is straightforward. The formulas may look a little scary, but
> the idea is not.


So I've been trying to figure this out. What is the relationship
between between the Fubini-Study metric and the inner product on its
vector space? I notice that the Fubini-Study metric has the same basic
format as the addition formula for tangents, which suggests that the
distance between two lines is simply the angle they subtend (and the
angle is determinined by the inner product). Then, for 3 collinear
lines, the sum of tangents formula agrees with the Fubini-Study metric.
Or at least I think it will agree. One is for differential
separations, the other is finitem, so I haven't checked that they
agree. And what if the lines are not collinear? I have no idea what
to do then. Is there a nice geometric picture for seeing the sum of
two angles that are not collinear?

So is this it? Is this what John Baez was alluding to? That the
"natural" distance between lines is angle? Seems natural enough to me.

Also, is there a straightforward way to arrive at the formula for the
Fubini-Study metric from the inner product on the vector space? The
only text I have that treats the matter just gives the Kähler form as
given, and gets the metric from that.

thanks


.

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