Where I am coming from: Similar to if you peel an orange, you cannot project the 2D surface of the Earth onto a 2D plane without curves. However, I’m pretty sure it’s possible to take the circumference of a circle and unwrap it into a line.
The following is my question broken down:
- Why is the transformation from a circular line that is 1D to a 1D line "easy"?
- Is there a word for this?
- Why is the transformation from a surface wrapped around a sphere (2D) to a (2D) plane without curves "hard"?
- Does this persist with $n$-dimensional hyperspheres to other $n$ dimensional elements?
I can't seem to describe my question properly; but when I mean elements, I want to refer to the basic element, if there is one, of an $n$th-dimension. For example, the "basic" element of the first dimension would be a line, and the second would be plane. So would it be right to extrapolate and assume that the "basic" element of the third dimension would be a cube?
- How would you describe a "basic" element of $\mathbb{R}^n$?
I know answers for this might include "it depends on the situation/it can be anything depending on context," so here it is: I would like a rigorous definition that connects a line, the mesh (I tried learning topology) of a square and a cube. I would also like if this could be extrapolated to higher dimensions so that my questions before make sense.
Question
Q. Is there a word for the difficulty of a mapping from a hyperspherical object in $\mathbb{R}^n$ to a "basic" element of $\mathbb{R}^n$; by which I hope to refer to a mesh that is defined in $\mathbb{R}^n$ with dimensions defined by independent perpendicular lines? Does this difficulty persist/change in higher dimensions?
Restriction: The type of transformations I described above should not involve infinite slices.
That is the sloppiest question I asked, so I apologize. I am not a specialist in mathematics such as topology, so an answer in layman's terms is what I am looking for. However, I can work with answers involving calculus and possibly a little bit of elementary real-analysis.
If I understand the qualitative intuition behind the question: The circle ($1$-sphere) is intrinsically flat, i.e., locally isometric to the Euclidean line. By contrast, if $n > 1$ then the $n$-sphere is not locally isometric to Euclidean $n$-space, essentially because Gaussian curvature and the Theorema Egregium.