# Questions tagged [isometric-immersion]

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### The hyperbolic plane $\mathbb{H}^2$ can't be isometrically immersed in $\mathbb{R}^3$ [closed]

It's easy to note that there is a local isometry between the hyperbolic plane $\mathbb{H}^2$ and the pseudosphere, since they have constant curvature equal to $-1$. Hilbert's Theorem.- There exists no ...
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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, ...
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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 ...
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### 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#...
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### 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 ...
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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 ...
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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 ...
148 views

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: $... 2answers 926 views ### Homotopy of space of immersions, Smale-Hirsch theorem If$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 ...