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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.

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