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# The Torus

*Not everything in life is simple*

Following college courses in differential geometry and general relativity, I became interested in calculating different quantities associated with the curvature of surfaces. The standard examples are the sphere and cylinder, but they're not very interesting; their curvature is constant. I undertook an exploration of a slightly more complex surface, the torus.

## A Model of the Torus

We begin with a torus with major radius *c* and minor radius *a*. We assume a nice torus, such that *c* > *a* > 0 (a *ring* torus).

We parameterize the torus's surface **x** in terms of coördinates *u* and *v*, each of which runs from 0 to 2π:

$$\mathbf{x}\left(u,v\right)=\left\{\begin{array}{l}x=\left(c+a\mathrm{cos}v\right)\mathrm{cos}u\\ y=\left(c+a\mathrm{cos}v\right)\mathrm{sin}u\\ z=a\mathrm{sin}v\end{array}\right.$$
The origin of the coördinate system lies on the outer equator, where *v* = 0.

## Results

I used both tensor calculus and differential geometry to calculate a variety of quantities related to curvature. The full details are available in a 19-page PDF, but these pages summarize the most important results:

**Curvature**: the torus's line element, metric, Christoffel symbols of the second kind, Riemann and Ricci tensors, and Ricci scalar.

**Geodesics**: the five families of geodesics on the torus, and some open questions.

**Shape Operator**: how the normal to the surface changes as we move on the surface.

**Parallel Transport**: how vectors rotate as they are parallel transported on the surface.

Note that these pages omit intermediate steps. The PDF contains full derivations.

All this, and I don't particularly like doughnuts.

Last updated 15 November 2005

`http://www.rdrop.com/~half/math/torus/index.xhtml`

All contents ©2004 Mark L. Irons

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