All Questions
9 questions
25
votes
1
answer
842
views
Alternate proofs that hyperbolic plane can’t be isometrically immersed in $\mathbb{R}^3$
A famous theorem of Hilbert says that there is no smooth immersion of the hyperbolic plane in 3-dimensional Euclidean space. The expositions of this that I know of (in eg do Carmo’s book on curves/...
- 251
13
votes
3
answers
5k
views
Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?
Consider the following question:
"Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometrically
immersed in three dimensional Eulidean space$(\mathbb{R}^3, g_{flat})$?"
I believe the answer to ...
- 3,245
5
votes
2
answers
207
views
Rozendorn's Article
I'm researching the isometric dips of the hyperbolic plane and in particular I'm interested in reading the results of Rozendorn who proved that the hyperbolic plane is isometrically immersed in $\...
- 143
5
votes
1
answer
431
views
Isometric immersion of $\mathbb H^2$ into $\mathbb R^\infty$ built by Bieberbach
I'm analyzing the following isometric immersion of $(\mathbb H^2,g_D)$ in $(\ell^2,g_\infty)$ given by $f(x,y)=(x_1,x_2,\dots,x_{2m-1},x_{2m},\dots)$ with
\begin{align}\label{5.1}
x_{2m-1}=\color{...
- 143
5
votes
1
answer
457
views
Can one use the continuity method to show that the two dimensional hyperbolic space can be immersed in five dimensional Euclidean space?
First of all, I must clarify at the outset that I am simply asking if there is an alternative way to solve an already known problem. It is known that the answer to my question is yes. The problem is ...
- 3,245
4
votes
1
answer
321
views
A local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$
I found this local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$, given by Schur (1886) in Über die Deformation der Räume constanten Riemannschen Krümmungsmaasses as follows, $(1\...
- 143
3
votes
0
answers
250
views
Immersion of a part of the hyperbolic plane in $\mathbb{R}^3$
I know that the pseudosphere is a regular surface with Gaussian curvature $-1$ that is not complete, also this surface is not complete. Hilbert's theorem ensures that there is no isometric immersion ...
- 143
1
vote
2
answers
266
views
Isometric embeddings of $\Bbb H^3$
Consider the upper-half space model of hyperbolic $3$-space $\Bbb H^{+}_{3}$, the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature ...
user515818
1
vote
0
answers
112
views
The best lower bound for isometric immersions
I just read Azov's article in the considered two classes of Riemannian metrics,
\begin{align*}
ds^2&=du_1^2+f(u_1)\sum_{i=2}^ldu_i,&f>0\\
ds^2&=g^2(u_1)\sum_{i=2}^ldu_i^2
,&g>0\...
- 143