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I was wondering if there exist an accepted name for an $n$-torus made by the product of hyperspheres $\mathbb{S}^d$, that is for the following set: $$ \mathbb{S}^d\times\stackrel{n}{\cdots}\times\mathbb{S}^d=\left\{(\mathbf{x}_1,\ldots,\mathbf{x}_n)\in\mathbb{R}^{n(d+1)}: \mathbf{x}_i=(x_{1,i},\ldots,x_{d+1,i}), ||\mathbf{x}_i||_2=1, i=1,\ldots,n \right\}. $$ Any topologists in the room?

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  • $\begingroup$ I'd say "generalized torus" and give the definintion you've given; I've never heard of another name, but perhaps someone else has. $\endgroup$ – John Hughes Sep 16 '14 at 15:22
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    $\begingroup$ Don't call it an $n$-torus. It's a product of spheres. $\endgroup$ – Qiaochu Yuan Sep 16 '14 at 15:24
  • $\begingroup$ @Travis, all right, so I will drop the "$n$-torus" terminology and go with "generalized torus" if there is not a more exact term available. $\endgroup$ – epsilone Sep 16 '14 at 15:41
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When $d = 1$, this is often called a flat ($n$-)torus. We say this because the metric induced on it by the ambient Euclidean space $\mathbb{R}^n$ is (locally) flat---by contrast, this is not the case for any $2$-torus embedded in $\mathbb{R}^3$. Note, though, that this is a statement about the metric geometry on the space, not the topology, where there is no such distinction.

For general $d$, one might call this an $n$-fold product of $d$-spheres. Anyway, as Qiaochu Yuan writes, you certainly shouldn't call this product an ($n$-)torus (when $d > 1$), as it would surely lead to confusion.

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