Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The standard embedded torus, parametrized by longitude $\alpha$ and latitude $\beta$ is given by $$ \begin{align} x & = (R+\cos\beta)\cos\alpha, \\ y & = (R+\cos\beta)\sin\alpha, \\ z & = \sin\beta. \end{align} $$ This is topologically the same as, but metrically different from, the flat torus $[0,2\pi]^2$, in which opposite sides are glued together and the distance between two points is the Euclidean length of the shortest path.

There is also tube $(\mathbb R\bmod 2\pi)\times C$ where $C$ is just an ordinary circle of unit radius. This is metrically the same as the flat torus if one uses the "intrinsic" metric, measuring distances along the surface, but one can also measure distances as lengths of chords passing through the interior. I.e., I want to regard this as embedded in $(\mathbb R\bmod2\pi)\times\mathbb R^2$ and consider the geometry of the way in which it's embedded rather than just the instrinsic metric.

My question is whether there is a standard name for this last object.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.