Isometric Immersion of the Torus

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Discussion Overview

The discussion revolves around finding an isometric immersion of the flat torus in \mathbb{R}^4, specifically focusing on demonstrating the injectivity of the differential of a proposed immersion function. Participants explore related concepts such as embeddings, Jacobians, and the implications of the immersion on the geometry of the torus.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant proposes the function f:\mathbb{S}^1\times\mathbb{S}^1\rightarrow \mathbb{R}^4 defined by f(\theta,\phi)=(\cos\theta,\sin\theta,\cos\phi, \sin \phi) as an isometric immersion of the flat torus.
  • Another participant provides the matrix of the differential of f and argues that it is injective by showing that the kernel is {0} due to the properties of sine and cosine functions.
  • A different participant suggests computing the Jacobian to prove that it has rank 2, indicating that the immersion is indeed an embedding.
  • This participant also notes that the immersion lies on a 3-sphere and mentions the relationship between the flat torus and a non-flat torus obtained through stereographic projection, highlighting the conformal mapping between them.
  • A question is raised about the existence of an isometric embedding of the flat Klein bottle in \mathbb{R}^4.

Areas of Agreement / Disagreement

Participants generally agree on the proposed immersion function and its properties, but there are differing views on the implications of the immersion and the relationship to other geometrical constructs, such as the non-flat torus and the Klein bottle.

Contextual Notes

The discussion includes assumptions about the properties of the functions involved and the nature of the embeddings, which may not be fully resolved. The implications of the stereographic projection and the nature of the torus in different geometrical contexts are also noted but not conclusively addressed.

felper
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Hello! I'm new here. I've been seeing this forum for a long time, but i never registered. I'd like to begin to contribute to this forum, i'll try it (even if my english doesn't help me).
So now, i ask you for help: I've to find an isometric immersion of the flat torus on \mathbb{R}^4. I know that the function f:\mathbb{S}^1\times\mathbb{S}^1\rightarrow \mathbb{R}^4 given by f(\theta,\phi)=(\cos\theta,\sin\theta,\cos\phi, \sin \phi ) is the indicated function, but i don't know how to demonstrate that it's differential is inyective.

Thanks for your help!
 
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Hi,

Notice that the matrix of the differential of f with respect to the (theta, phi) coordinates on T² and the standard coordinates on R^4 is the 4x2 matrix

-sin(theta) 0
cos(theta) 0
0 -sin(phi)
0 cos(phi)

Since cos(x) and sin(x) never vanish simultaneously, it is easy to see that the linear map associated with this matrix has kernel={0} and so is injective.
 
felper said:
Hello! I'm new here. I've been seeing this forum for a long time, but i never registered. I'd like to begin to contribute to this forum, i'll try it (even if my english doesn't help me).
So now, i ask you for help: I've to find an isometric immersion of the flat torus on \mathbb{R}^4. I know that the function f:\mathbb{S}^1\times\mathbb{S}^1\rightarrow \mathbb{R}^4 given by f(\theta,\phi)=(\cos\theta,\sin\theta,\cos\phi, \sin \phi ) is the indicated function, but i don't know how to demonstrate that it's differential is inyective.

Thanks for your help!

just compute the Jacobian and prove that it has rank 2.

BTW: this is more than an immersion. It is an embedding. Further the length of every point considered as a vector in R^4 is constant so this torus lies on a 3 sphere. Try computing the equation of the torus in R^3 obtained from this one by stereographic projection. This new torus is not flat yet the mapping between it and the flat torus is conformal.

Your immersion is of the Euclidean plane into R^4.

Question: Is their an isometric embedding of the flat Klein bottle in R^4?
 
Last edited:
Thanks, i could demonstrate it. I'll think about the question.
 

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