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.