# Questions tagged [isometric-immersion]

The isometric-immersion tag has no usage guidance.

15
questions

**-1**

votes

**0**answers

183 views

### 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 ...

**23**

votes

**1**answer

483 views

### 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/...

**8**

votes

**0**answers

110 views

### 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 ...

**8**

votes

**1**answer

151 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#...

**4**

votes

**1**answer

208 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 ...

**5**

votes

**0**answers

121 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 ...

**3**

votes

**0**answers

93 views

### 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 $...

**8**

votes

**2**answers

347 views

### Is there a $W^{2,2}$ isometric embedding of the flat torus into $\mathbb{R}^3$?

It is well known that there exists a $C^1$ isometric embedding of flat torus into $\mathbb{R}^3$, and that this embedding cannot be $C^2$.
Is there a $W^{2,2}$ isometric embedding? (i.e an isometric ...

**2**

votes

**1**answer

116 views

### Preserving connections immersions

I am dealing with Riemannian immersions and I am stuck on the following: Given a totally geodesic immerserd surface $S$ on a compact riemannian manifold $M$ with metric $g$, is there another metric $\...

**1**

vote

**0**answers

65 views

### 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 ...

**6**

votes

**0**answers

438 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, ...

**5**

votes

**2**answers

148 views

### Are all symmetries of the Dirichlet functional isometries?

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: $...

**10**

votes

**2**answers

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>...

**10**

votes

**3**answers

3k views

### Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?

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 ...

**5**

votes

**1**answer

401 views

### Can one use the continuity method to show that the two dimensional hyperbolic space can be immersed in five dimensional Euclidean space?

First of all, I must clarify at the outset that I am simply asking if there is an alternative way to solve an already known problem. It is known that the answer to my question is yes. The problem is ...