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I am interested in the topological transformations that map a square to other surfaces, such as a torus and a sphere, particularly in the context of taking the Jacobian of these transformations.

For the square-to-torus transformation, the identification mapping

ϕ:(α,β)↦((sin(2πα),cos(2πα)),(sin(2πβ),cos(2πβ)))

In this case, taking the Jacobian matrix of ϕ can provide us with information on how the local "stretching" or "compression" occurs when the square is mapped onto the torus. What is the significance or interpretation of the Jacobian in this context? Are there any applications where this Jacobian could be particularly useful?

Square-to-Sphere Transformation I am also curious if a similar kind of mapping exists for transforming a square into a sphere via adjacent edge identification, specifically a mapping that lends itself to Jacobian analysis.

  • Is there a well-defined transformation from a square to a sphere, akin to the square-to-torus transformation, where the Jacobian can be taken?
  • If such a transformation exists, what could be the interpretation or application of its Jacobian?

Any insights or references would be greatly appreciated.

Dwip Dalal
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  • Think spherical coordinates. – Ted Shifrin Oct 05 '23 at 02:44
  • Would the following be the correct thinking?

    x, y, z​=sin(θ)cos(ϕ), sin(θ)sin(ϕ), cos(θ)​ and now taking jacobian of this?

     – Dwip Dalal Oct 05 '23 at 02:51
  • Since in this case, we are performing adjacent edge identification, so I am worried if this would be the right way of calculating Jacobian? – Dwip Dalal Oct 05 '23 at 02:53
  • No, why adjacent edge identification? That is not necessary. Are you going to identify all four edges to a point? And what is your point in doing this? – Ted Shifrin Oct 05 '23 at 03:16
  • Let a1, a2, a3, and a4 be four edges of the square. If we identify a1 with a2 and a3 with a4, then we would get a sphere, right? I have probability values defined on the square in R2, and now I want to transfer these probability values to the sphere. For doing this, I think we would need Jacobian of this transformation and so I am wondering how to calculate it. – Dwip Dalal Oct 05 '23 at 03:23
  • Due to the nature of my problem statement - edge identification is sort of necessary for me to use for constructing the sphere from a square. Hence I need to find the jacobian of this transformation. I hope I am thinking in the right direction? – Dwip Dalal Oct 05 '23 at 03:29
  • I still don’t get it. But yes, you can do it with adjacent edge identification. Here’s a hint: Stand the square vertically (like a diamond). Identify the top edges and identify the bottom edges. Think of a double cone with vertices at the top and bottom. You can work out formulas. – Ted Shifrin Oct 05 '23 at 03:53

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