Questions tagged [isometric-immersion]
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31 questions
6
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229
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Reference request: Moduli space of immersed polygons
I am looking for references describing the structure of the set of immersions of the disk $D^2$ in the Euclidean plane $\mathbb{R}^2$ or in the hyperbolic plane $\mathbb{H}^2$ such that the boundary ...
- 687
2
votes
0
answers
114
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Comparison between the topology of the image of an immersion and of its domain
Let $M^n$ be a closed $n$-dimensional manifold and $f:M^n\to \mathbb{R}^d$ an immersion. Is it possible to compare the homologies of $M^n$ and the one of $f(M^n)$? For instance, is it possible to ...
- 687
5
votes
0
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192
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What's the lowest-dimensional Euclidean space in which a hexagonal 2-torus smoothly embeds isometrically?
Here a hexagonal 2-torus TH means any Riemannian flat torus obtained by identifying opposite edges of a regular hexagon in the plane.
It's easy to see that TH can be smoothly (C��) isometrically ...
- 3,798
5
votes
0
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250
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Flat minimal torus in a sphere
Suppose that a flat $n$-dimensional torus $T$ is isometrically embedded into the unit sphere $\mathbb{S}^N$ of large dimension.
Furthermore, $T$ is a minimal submanifold in $\mathbb{S}^N$.
Is it true ...
- 47.3k
2
votes
1
answer
197
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Smooth isometric immersions of the a hemisphere in $\mathbb R^3$
Let $S$ be a disc endowed with a spherical metric (constant positive Gaussian curvature). What is known about its smooth isometric immersions in $\mathbb R^3$?
By [GS20], an immersion is uniquely ...
- 857
1
vote
2
answers
411
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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
2
votes
2
answers
358
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A property for maps between metric spaces
Let $X, Y$ be metric spaces with distance functions denoted by $d_X, d_Y$ respectively. Consider a map $f \colon X \rightarrow Y$. I am interested in the following property: for every $x,y,z \in X$, ...
- 409
4
votes
0
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121
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Are Sobolev isometries in Minkowski space smooth
Let $\Omega\subset\mathbb{R}^d$ be an open regular domain and let $f\in W^{1,\infty}(\Omega;\mathbb{R}^d)$ satisfy that $df\in\operatorname{SO}(d)$ almost-everywhere. It was proved by Reshetnyak (in a ...
- 857
5
votes
1
answer
306
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Approximate isometric embeddings of surfaces
The fundamental theorem of surfaces states that if symmetric matrices $g_{ij}$, $l_{ij}\colon U\subset R^2\to R$, where $U$ is open and $g_{ij}$ is positive definite satisfy the Gauss and Codazzi ...
- 7,434
7
votes
2
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348
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Are polyhedra with equilateral triangular faces rigid?
Convex polyhedra are rigid by Cauchy’s theorem. Steffen’s polyhedron is an example of a non-convex polyhedron that is flexible (i.e., non-rigid). However, it appears to have edges of different lengths....
- 264
3
votes
0
answers
135
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Isometric embedding of a 2-dimensional orbifold with constant curvature and three cone points
There are classical surfaces of revolution, shaped like footballs, that have constant positive curvature, except for their two cone points. How about such a surface with three cone points?
To give ...
- 6,553
5
votes
1
answer
491
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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
1
vote
0
answers
133
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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
4
votes
1
answer
385
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
5
votes
2
answers
264
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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