From: jeffcoat@stat.rice.edu (Mark Evans Jeffcoat)
Subject: Re: Immersion Theory
Date: 13 May 1999 16:12:44 GMT
Newsgroups: sci.math.research
Keywords: Isometric immersion of H^2 into R^4?
Roger Arbogast (roar0000@stud.uni-sb.de) wrote:
: Hi everybody! I am studying mathematics at an university in Germany and my
: question is the following: Does anybody know if there is an isometric
: immersion of the two- dimensional Hyperbolic Space (i.e. the Hyperbolic
: Plane) into the four- dimensional (n=4) Euclidean Space (i.e. IR^4)? I know
: that the answer is yes if n=5 and that the answer is no if n=3. I would be
: very thankful for a hint to any literature for the case n=4.
: Thanks Roger (email: roar0000@stud.uni-sb.de)!
:
According to Do Carmo, _Differential Geometry of Curves and Surfaces_,
this is still an open question. (The edition I have was printed
in 1976, so any solution that might exist ought to be quite
recent.)
==============================================================================
From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: embedding the hyperbolic plane?
Date: 12 Jul 1999 12:30:10 -0400
Newsgroups: sci.math
Pertti Lounesto writes:
>Dave Rusin wrote:
>
>> Nick Halloway wrote:
>> >Can the hyperbolic plane be embedded isometrically in 4-space?
>
>I do not understand the question. Please, explain.
"The hyperbolic plane" can be taken to mean "the smooth manifold
R^2, equipped with a complete Riemannian metric of constant
curvature -1" (there is a unique such metric on R^2). "Embedded
isometrically in 4-space" must mean, here, "embedded as a smooth,
closed submanifold of R^4 in such a way that the Riemannian metric
induced from the flat metric of R^4 is the given metric on R^2".
>> Unknown (!). There's an embedding into R^5, none into R^3.
>
>The answer does not give me glue, to anything. Please, expand.
John Klein sometimes posts here. He, or someone else who is
familiar with the work of Gromov, might comment on whether
it's known if there is a *non-smooth* embedding of R^2 as
a closed subset of R^4, such that (the non-Riemannian)
metric induced on R^2 (in which the distance between two
points of R^2 is the infimum of the lengths of curves on the
image in R^4 joining the images of the points) is isometric
(in the sense of metric spaces) to the hyperbolic metric
on R^2.
Lee Rudolph
==============================================================================
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: embedding the hyperbolic plane?
Date: 12 Jul 1999 23:03:32 GMT
Newsgroups: sci.math
In article <3788CBDD.7F46FBD6@hut.fi>,
Pertti Lounesto wrote:
>
>Dave Rusin wrote:
>
>> Nick Halloway wrote:
>> >Can the hyperbolic plane be embedded isometrically in 4-space?
>>
>> Unknown (!). There's an embedding into R^5, none into R^3.
>
>Can you give some details, both of the embedding into 5D and the
>unknowledgeability of an embedding into 4D.
Sorry, not really. This is recent work which only crossed my desk by
reference. Some Math Reviews citations are below. I didn't say an
embedding into R^4 was "unknowable", rather, that it simply isn't yet
known. If I had to bet, I'd guess there is no embedding into R^4, but that
we haven't yet found the tools to show this.
Just to clarify the problem here, the hyperbolic plane can be thought of
as the topological space usually known as the open unit disk in R^2.
Clearly, it can be embedded into R^2 ! But in addition to the topological
structure, it can be given the structure of a Riemannian manifold, that is,
we can give a method for computing distances on it _distinct from_ the
usual metric. (Ultimately this metric comes from an inner product on each
tangent plane T_p(M) of the manifold; the hyperbolic plane is a space
in which the inner product is of signature (+1, -1). ) An isometric
embedding would then be one in which distances between points are
preserved.
I'm embarassed to say I cannot recall quite why there is no embedding of
H^2 into R^3. It isn't quite the constant negative curvature which
prohibits this, as there _is_ such a surface in R^3 which is
homeomorphic not to the disk but to the punctured disk. It looks rather
like a bugle horn with Gabriel infinitely far away...
By the way, you can _always_ embed a Riemannian manifold into _some_
Euclidean space; that's the Nash embedding theorem. See e.g.
http://www.math-atlas.org/index/57RXX.html
dave
96e:53098 53C42 (53A35)
Oláh-Gál, Róbert
The $n$-dimensional hyperbolic space in ${E}\sp {4n-3}$. (English.
English summary)
Publ. Math. Debrecen 46 (1995), no. 3-4, 205--213.
D. Blanusa constructed an isometric immersion of class $C\sp \infty$
of the $n$-dimensional hyperbolic space $H\sp n$ into the Euclidean
space $E\sp {6n-5}$ [Monatsh. Math. 59 (1955), 217--229; MR 17, 188f].
By modifying this construction, the author constructs an isometric
immersion of class $C\sp \infty$ of the $n$-dimensional hyperbolic
space $H\sp n$ into the Euclidean space $E\sp {4n-3}$.
Reviewed by Vladislav Goldberg
_________________________________________________________________
96e:53084 53C42 (53A35)
Azov, D. G.(RS-CHLT)
Isometric embedding of $n$-dimensional metrics into Euclidean and
spherical spaces. (Russian. Russian summary)
Vestnik Chelyabinsk. Univ. Ser. 3 Mat. Mekh. 1994, no. 1, 12--16.
D. Blanusa constructed an isometric immersion of the $n$-dimensional
hyperbolic space $H\sp n$ into the Euclidean space $E\sp {6n-5}$
[Monatsh. Math. 59 (1955), 217--229; MR 17, 188f] and into the
spherical space $S\sp {6n-4}$ [see Glasnik Mat.-Fiz. Astronom. Ser. II
Drustvo Mat. Fiz. Hrvatske 19 (1964), 53--61; MR 30 #2421]. E. R.
Rozendorn [Akad. Nauk Armyan. SSR Dokl. 30 (1960), 197--199; MR
24\#A3568] constructed an embedding of the metric $ds\sp 2 = du\sp 2 +
f\sp 2 (u) dv\sp 2$ into the Euclidean space $E\sp 5$. In the paper
under review, the author considers two classes of Riemannian metrics:
$ds\sp 2 = du\sb 1\sp 2 + f\sp 2 (u\sb 1) \sum\sb {i=2}\sp n du\sb
i\sp 2$, $f > 0$, and $ds\sp 2 = g\sp 2 (u\sb 1) \sum\sb {i=2}\sp n
du\sb i\sp 2$, $g > 0$, and proves that if $n > 2$, then these two
metrics admit an isometric embedding into the Euclidean space $E\sp
{4n-4}$ and into the spherical space $S\sp {4n-4}$, and if $n = 2$,
then these two metrics admit an isometric embedding into $E\sp 4$ and
$S\sp 4$ in the form of surfaces with singular lines. The class $C\sp
m$ of all these embeddings coincides with the class of the functions
$f$ and $g$. The author indicates that the isometric embedding of the
Lobachevskii plane into $E\sp 4$ was obtained earlier by I. Kh.
Sabitov [Sibirsk. Mat. Zh. 30 (1989), no. 5, 179--186, 218; MR
90k:53099].
Reviewed by Vladislav Goldberg
_________________________________________________________________
90k:53099 53C42 (53A35)
Sabitov, I. Kh.
Isometric immersions of the Lobachevski\u\i plane in $E\sp 4$.
(Russian) Sibirsk. Mat. Zh. 30 (1989), no. 5, 179--186, 218;
translation in Siberian Math. J. 30 (1989), no. 5, 805--811
It is well known that the Lobachevskii plane can be isometrically
imbedded in $E\sp 6$ or immersed in $E\sp 5$ in the class of $C\sp
\infty$-surfaces. In this paper, the author proves the following
result: The Lobachevskii plane can be isometrically immersed in $E\sp
4$ in the form of a piecewise-analytic surface with smoothness in the
class $C\sp {0,1}$. Moreover, the Lobachevskii plane cannot be
isometrically immersed in $E\sp 4$ as a generalized $C\sp 1$-surface
of revolution: $x\sb 1+ix\sb 2=F(u)e\sp {iPv}$, $x\sb 3+ix\sb
4=G(u)e\sp {iQv}$.
Reviewed by Yi Bing Shen
© Copyright American Mathematical Society 1990, 1999
==============================================================================
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: embedding the hyperbolic plane?
Date: 16 Jul 1999 15:58:34 GMT
Newsgroups: sci.math
"Nick Halloway" wrote:
>Can the hyperbolic plane be embedded isometrically in 4-space?
Hoping to address a few of the finer points in the recent discussion of the
hyperbolic disk, I looked up the article by Olah-Gal whose review I
cited in another post. It contains a nice summary of the situation in just a
few pages. I won't display the whole article of course, but I will
quote sections which refer to our previous discussion.
1. The "best" embedding of the hyperbolic plane into Euclidean space seems
to be a construction of Blanusa (Monatshefte Math. 59 (1955) 217-229)
which is an explicit, smooth (C-infinity) embedding of the plane onto a
subset of R^6 which preserves the hyperbolic metric. I will give the
formulas below.
2. Olah-Gal notes in his article that by deleting the last coordinate
(i.e. projecting to R^5) we obtain a surface contained in a smaller
space, which still has constant negative curvature. However, this
projection is not one-to-one (it is only a local embedding) and has
a singular point at the origin. A handy illustration is provided in the
article showing an analogous situation in which a surface in R^4
[drawn on paper on my 2-dimensional desk...] projects to a surface in R^3.
3. From the introduction: "[Blanusa's map] is of class C^\infty, but it is not
analytic. A construction of an analytical embedding of the hyperbolic plane
into E^n (with sufficiently large n) is unknown even these days." [1995]
I have a reference to Gromov's "Partial Differential Relations"
which indicates he proves there that hyperbolic k-space can be
analytically isometrically _immersed_ into Euclidean (5k-5)-space.
I don't know for certain that this applies to k=2.
4. From p.208: "In 1955 Amsler [Math. Ann. 130, 234-256] proved that each
surface of E^3 with constant negative curvature has an edge (i.e. it
contains a curve consisting of singularities), and showed the nonexistence
in E^3 of surfaces of constant negative curvature with singularity
consisting of one point alone."
In a previous post I described a surface called
"Gabriel's horn" of constant negative curvature, with a singular curve
around the bell of the horn. Another poster noted that the non-immersibility
of H^2 in R^3 had something to do with singularities. I guess it was Hilbert
who first showed H^2 does not embed (or even immerse) in R^3.
The situation in R^4 is open as far as I can tell. Any _compact_
domain in the hyperbolic plane can be isometrically immersed into R^4;
see Gromov's book. (Indeed, from all reports, one should see Gromov's book
for just about everything regarding smoothly embedding Riemannian manifolds.)
Olah-Gal's article goes on to describe the constructions in substantial
detail and to comment on generalizations.
Now I will describe the embedding of R^2 into R^6. This is not exactly
what one would call an "obvious" construction, but there are certain
patterns to it which suggest how Blanusa might have been led to it.
We will send the point (u,v) to a point with six coordinates (x1, ..., x6)
which are functions of u and v of the special forms
x1 = x1(u)
x2 = f1(u) sin( v psi1(u) )
x3 = f1(u) cos( v psi1(u) )
x4 = f2(u) sin( v psi2(u) )
x5 = f2(u) cos( v psi2(u) )
x6 = v
I will describe the functions x1, f1, f2, psi1, psi2 (of one variable each)
in stages.
Let [x] denote the integer part of x.
The functions psi1 and psi2 are periodic functions of |u| (period = 2)
which on [0,2] are exponentials of linear maps:
psi1(u) = exp( 2*[ (|u|+1)/2 ] + 5 )
psi2(u) = exp( 2*[ ( |u| )/2 ] + 6 )
There are discontinuities in the psi_i at certain integers but other
parts of the construction will keep the x_i smooth.
Define two functions phi_i via certain normalized antiderivatives:
writing F(x) = sin( pi x )/exp( sin^{-2}(pi x) ) we have
phi1(u) = { (1/A) integral( F(x), x=0 to x= u+1 ) }^(1/2)
phi2(u) = { (1/A) integral( F(x), x=0 to x= u ) }^(1/2)
where
A = integral( F(x), x=0 to x=1 ) = 0.141327...
These functions phi_i are non-negative, periodic, and satisfy
phi1^2 + phi2^2 = 1 and phi1(u) = phi2(u+1). You can think of the phi_i
as being very smooth versions of |sin(pi u)| and |cos(pi u)|.
Now set f_i(u) = sinh(u) phi_i(u)/psi_i(u) for i=1,2 and
define x1 to be an antiderivative of 1-(f1')^2-(f2')^2 having x1(0)=0.
Loosely speaking the mapping x1 sends lines in R^2 far enough away,
and the the coordinates x2, x3, x4, x5 allow the points in these line
to spin around in four perpendicular directions, with enough spinning
to account for the fact that the images of lines are supposed to grow
very long. The last coordinate x6 merely adds a motion in another
perpendicular direction to separate points so that these curves don't
self-intersect.
The metric which R^2 inherits from this embedding into R^6 comes out to
ds^2 = du^2 + cosh(u)^2 dv^2, from which one finds the curvature to be
constant and negative, making R^2 into the hyperbolic plane.
dave
==============================================================================
[additional related review -- djr]
80c:53071 53C40
Yang, Paul
Curvatures of complex submanifolds of ${C}\sp{n}$.
J. Differential Geom. 12 (1977), no. 4, 499--511 (1978).
The author treats the following question: Does there exist a complete
complex submanifold of complex space with holomorphic sectional
curvature bounded away from zero? The motivation for the question is
due to the fact proved by S. Bochner (Bull. Amer. Math. Soc. 53
(1947), 179 - 195; MR 8, 490) that the Poincare metric of constant
negative curvature on the unit disc cannot be holomorphically embedded
in complex Euclidean space even locally. The author gives partial
results to the above question. The first claim is that a negative
answer to this question would imply that there is no bounded complete
submanifold of complex space. Secondly, the question is considered for
the hypersurfaces of complex space. Namely, the fact that complete
complex hypersurfaces of complex space cannot have strongly negative
holomorphic sectional curvatures is proved. Thirdly, the author shows
that for arbitrary codimension this question can be reduced to the
consideration of holomorphic curves. Using the higher order curvature
functions, the author shows that two such functions are enough to
determine a holomorphic curve uniquely up to a rigid motion in complex
space, thus providing a justification for the generalization of the
fact for hypersurfaces. Fourthly, the author derives a criterion,
involving the curvature behavior at infinity of a simply connected
metric Riemannian surface, for it to be confor- mally equivalent to
the disc, which is a complement to results of R. E. Greene and H.-H.
Wu (ibid. 77 (1971), 1045 - 1049; MR 44 #473). Lastly, a curvature
estimate is proved for a piece of a curve in complex 2-space which is
a graph over a domain in complex line.
Reviewed by Yoshiaki Maeda
Cited in reviews: 80j:53063
© Copyright American Mathematical Society 1980, 1999