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Re: Euclidean models of hyperbolic 2-manifolds
Posted:
Apr 17, 2000 3:43 PM
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Roger wrote:
> "R. Bryant" wrote
[in response to a question from James Propp.-RLB]
> > Yes. What you are asking for is that there be a locally homogeneous
> > embedding of M into R^n for some n. Now, a locally homogeneous
> > immersion of M into R^n does not exist, even locally. The reason
> > is, ultimately, that this would force a nontrivial homomorphism of
> > SL(2,R) into the Euclidean motion group, which clearly does not
> > exist.
>
> This problem disappears if n is infinite. Does anyone know if
> there is a nice embedding of hyperbolic space into Hilbert
> space? Ie, one that has simple formulas, preserves distance,
> and exhibits its symmetries as Euclidean motions?
You are quite right. There are many isometric embeddings of
the hyperbolic plane into a Hilbert space that are equivariant
with respect to an action of the isometry group of the hyperbolic
plane. For example, let |dA| be the usual SL(2,R)-invariant
area measure on the hyperbolic plane and let L^2(|dA|) denote
the space of L^2 functions on SL(2,R) with respect to this
measure. Let d(p,q) be the hyperbolic distance between p
and q in the hyperbolic plane and let f_q(p) = exp(-d(p,q)^2).
Then the map that sends q to f_q (which lies in L^2(|dA|))
is an embedding that is equivariant with respect to the
isometry group of the hyperbolic plane (the action on L^2(|dA|)
is via pullback). This alone implies that this map embeds
the hyperbolic plane into L^2(|dA|) in such a way that the
induced metric is some constant multiple of the hyperbolic
metric. Thus, after scaling the embedding by a constant,
if necessary, you get an equivariant isometric embedding.
Of course, there are many variations on this theme, but you
get the idea.
One amplification of my earlier response that you may find
of interest is this: It is not possible to (smoothly) isometrically
embed the hyperbolic plane into R^n (n finite, of course) so that
it is equivariant with respect to even a one-parameter subgroup
of the isometry group of the hyperbolic plane. Thus, one
cannot even preserve one dimension's worth of symmetry in
any given isometric embedding. This impossibility basically
follows from the exponential volume growth of geodesic
disks, but I don't see how to give a rigorous proof along those
lines. (I'm sure this is in the literature somewhere, but I don't
know where to find it. The proof that I have in mind is kind of
ugly, even though it's not long.)
Yours,
Robert Bryant
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