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Results tagged with riemannian-geometry
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user 171387
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
5
votes
1
answer
428
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
Accepted
Isometric immersion of $\mathbb H^2$ into $\mathbb R^\infty$ built by Bieberbach
After a few tries I got the following:
Instead of taking real variable I take complex variable, that is let $z_m=\dfrac{\color{red}{2}z^m}{\sqrt{m}}$, donde $z_m=x_{2m-1}+ix_{2m}$. Then $dz_m=\color{r …
- 143
1
vote
0
answers
108
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\end{align*}
and solved …
- 143
4
votes
1
answer
318
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\le …
- 236
5
votes
2
answers
205
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 $\math …
- 346
3
votes
0
answers
238
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 o …
- 143
2
votes
0
answers
332
views
Question in the proof of Hilbert's theorem
I'm researching about Hilbert's theorem which says that there isn't isometric immersion of a complete surface with constant negative Gaussian curvature in $\mathbb{R}^3$. I'm taking as a reference the …
- 143