From: Robert Bryant
Newsgroups: sci.math.research
Subject: Re: embedding hyperbolic space in Euclidean space
Date: Mon, 23 Nov 1998 09:22:14 -0500
Jim Propp wrote:
>
> What is the smallest n such that hyperbolic k-space can be embedded
> isometrically in Euclidean n-space? (I've heard that the answer is
> that it can't be done for ANY n, because of the growth at infinity.)
Actually, by the Nash Embedding Theorem (proved while Nash
was at MIT, by the way), any Riemannian manifold can be isometrically
embedded in some Euclidean space. This general result can be
sharpened in various ways. You can look at Greene's Memoirs of
the AMS, No. 97, for some more information. He proves that
a Riemannian manifold of dimension k can be isometrically
embedded in Eudclidean (2k+1)(6k+14)-space. In Gromov's
'Partial Differential Relations', he proves that you can
isometrically imbed into (k+2)(k+3)/2, the best known
general bound.
For the specific case of hyperbolic k-space, the best
bound is not known, but it is certainly lower than quadratic.
In PDR, Gromov proves that hyperbolic k-space can be
analytically isometrically immersed into Euclidean
(5k-5)-space (see the Corollary on p. 296).
As you probably know, Hilbert proved that the hyperbolic
plane cannot even be immersed isometrically in Euclidean
3-space. Gromov showed (PDR, p. 294) that any compact
domain in the hyperbolic plane can be isometrically immersed
into any open subset of Euclidean 4-space. Whether it can
be done globally or not, I don't know.
> What is the smallest n such that any compact subset of hyperbolic
> k-space can be embedded isometrically in Euclidean n-space?
> (I've heard that the answer is 2n-1).
Well, E. Cartan proved long ago that if an k-manifold in
Euclidean n-space had constant negative sectional curvature,
then n is at least 2k-1. He also showed that there were many
local examples of k-manifolds of constant negative curvature
in Euclidean 2k-1 space. (In fact, they are quite flexible,
depending on k(k-1) functions of 1 variable in Cartan's sense.)
However, except in the case k=2, the analog of Hilbert's
theorem is not known, i.e., when k>2 is it not known
whether hyperbolic k-space can be isometrically immersed
(let alone embedded) into Euclidean 2k-1 space.
By the way, Hilbert's proof actually shows that there is
a finite upper bound on the area of a domain in the hyperbolic
plane that can be isometrically immersed in Euclidean 3-space,
so if your question it interpreted literally, you have
to go higher than 2k-1 even when k=2.
Yours,
Robert Bryant