Questions tagged [isometric-immersion]
The isometric-immersion tag has no usage guidance.
8
questions with no upvoted or accepted answers
6
votes
0
answers
595
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Properties that are between intrinsic and extrinsic in Riemannian geometry
Motivation (update): I am interested in properties/structures/objects that are determined by the metric alone, but are not among the usual ones that we call intrinsic, like Levi-Civita connection, ...
5
votes
0
answers
132
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"Inflating" a closed, defined metric, manifold
Consider a two-dimensional Riemannian manifold homeomorphic to the sphere, with a defined metric.
Since we do not suppose that manifold to have a positive curvature,
we are not in the hypotheses of ...
4
votes
0
answers
89
views
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 ...
3
votes
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83
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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 ...
3
votes
0
answers
222
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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 ...
3
votes
0
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100
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Is there a weak isometric completion to a $W^{2,2}$ isometric immersion?
Let $g$ be a smooth Riemannian metric on $\mathbb{R}^d$.
Let $D=D^k \subseteq \mathbb{R}^d$ be the $k$-dimensional closed unit disk. ($k<d$).
Suppose we are given a $W^{2,2}$ isometric immersion $...
1
vote
0
answers
107
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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\...
1
vote
0
answers
315
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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 ...