# Questions tagged [isometric-immersion]

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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 ...
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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 ...
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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....
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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 ...
421 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{...
1 vote
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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\...
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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 ...
1 vote
301 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 ...
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### 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/...
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### Every immersion can be deformed to have only transverse self-intersections

I asked this question some time ago in MSE, but obtained no answer. Maybe this is the right place to post it. Let $f : M^n \to \overline{M}^{n+k}$ be an immersion between smooth manifolds. Is it true ...
209 views

### Question on Nash's paper on $C^1$ isometric immersions: Why approximating the error tensor $\delta$?

I am trying to go through the classical paper by Nash on the existance of $C^1$ isometric immersion of a Riemannian manifold $(M,g)$ (here is the Jstor link: https://www.jstor.org/stable/1969840?seq=1#...
287 views

### Bending the hemisphere

Let $S^+$ be the upper hemisphere of the standard sphere in $\mathbb R^3$ and $b$ -- the boundary of $S^+$ (the equator). Let $b'$ be a small isometric deformation of $b$ (a nearby curve of the same ...
131 views

### "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 ...
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1 vote
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### Characterizing a surface in $R^3$ with a given metric [closed]

Let $g$ be a Riemannian metric in $\mathbb{R}^2$. How can I find a surface in $R^3$ such that their curvature are the same? The shape of the surface in $3$ space is important. I mean I dont like to ...
551 views

### 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, ...
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This is a cross-post from MSE (no answer there). Let $M,N$ be oriented $d$-dimensional Riemannian manifolds, $M$ compact*, and let $f:M \to N$ be smooth. Consider the Dirichlet energy functional: $... 11 votes 2 answers 1k views ### Homotopy of space of immersions, Smale-Hirsch theorem If$M$and$N$are manifolds with$\dim M< \dim N$, we denote by$Imm\left(M,N\right)$the space of immersions of$M$in$N$. Let$M$and$M'$be a simply connected manifolds of dimensions$m>...
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 ...