All isometries of the n-torus

  • Thread starter Palindrom
  • Start date
In summary, the isometry group on any Riemannian manifold M is a group of isometries, with specific forms for R^n and S^n. One possible guess for the isometry group of T^n is O(1) X ... X O(1) with the product group structure, but this may not work. Another approach could be using the fact that T^n is homeomorphic to R^n/Z^n, but there may be issues with lifting isometries from the torus to the plane. The textbook by Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, may provide helpful information.
  • #1
Palindrom
263
0
How does one start this kind of question? I'm completely stumped.
 
Physics news on Phys.org
  • #2
The set of all isometries on any riemannian manifold M forms a group called the isometry group on M. For R^n this is the set of all rigid motions each of which has the form f(x) = Ax + b where A is in O(n) and b is in R^n. For S^n this is the set of all orthagonal transformations, i.e. O(n). Now here is my guess. Since T^n = S^1 X ... X S^1 (n times) then perhaps it's isometry group is O(1) X ... X O(1) with the product group structure. If this fails perhaps you could use the fact that T^n is homeomorphic to R^n/Z^n. Good luck
 
  • #3
First of all, thanks a lot for taking the time.

I'd already figured out that T^n being isometric to R^n/Z^n might be helpful. But it is true that in this case, every isometry of R^n/Z^n extends to an isometry of R^n?

This is actually what's bothering me.

I'll admit I hadn't thought of the S^1 X...X S^1 idea, but it looks like I have a similar problem in this case.
 
  • #4
Lifting from torus to plane

Palindrom said:
I'd already figured out that T^n being isometric to R^n/Z^n might be helpful. But it is true that in this case, every isometry of R^n/Z^n extends to an isometry of R^n?

"Lift", not "extend". The textbook by Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry should help.

Chris Hillman
 

1. What is an isometry of the n-torus?

An isometry of the n-torus is a transformation that preserves distances and angles between points on the n-torus. In simpler terms, it is a mapping that does not change the shape or size of the n-torus.

2. How many isometries does the n-torus have?

The number of isometries of the n-torus depends on the dimension of the torus. For a 2-dimensional torus, there are infinitely many isometries, while for higher dimensions, there are a finite number of isometries.

3. What are the different types of isometries of the n-torus?

There are three types of isometries of the n-torus: translations, rotations, and reflections. Translations involve sliding the torus in a certain direction without changing its shape, rotations involve rotating the torus around an axis, and reflections involve flipping the torus across a plane.

4. How are isometries of the n-torus useful in mathematics?

Isometries of the n-torus are useful in mathematics because they allow us to study and understand the properties of geometric objects without having to explicitly work with their coordinates. They also help in visualizing and understanding abstract concepts such as symmetry and group theory.

5. Are all isometries of the n-torus unique?

No, not all isometries of the n-torus are unique. There can be multiple isometries that result in the same transformation of the torus. For example, a rotation of 180 degrees around the center of a 2-dimensional torus can also be achieved by two successive reflections across different planes.

Similar threads

  • Differential Geometry
Replies
2
Views
1K
  • Differential Geometry
Replies
1
Views
1K
  • Differential Geometry
Replies
16
Views
2K
  • Sci-Fi Writing and World Building
Replies
9
Views
2K
  • Differential Geometry
Replies
4
Views
4K
Replies
2
Views
2K
Replies
4
Views
253
  • Topology and Analysis
Replies
2
Views
977
Replies
2
Views
6K
  • Differential Geometry
Replies
4
Views
5K
Back
Top