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A torus is the set of points in $\mathbb{R}^3$ at a distance $b$ from the circle of radius $a$.

I want to know why if $a=b$ or $b>a$ then the "torus" given is not a smooth manifold. I can figure it out intuitively speaking, but I want a formal argument.

EQJ
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    Draw a picture. If b>a there is a clear self intersection, if b=a there is a cusp at the origin. –  Aug 18 '16 at 00:51
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    There is a way to say what you are pointing out in a formal language? – EQJ Aug 18 '16 at 01:18
  • Is the question here: "Why is this not a torus, as in diffeomorphic to $S^1\times S^1$?" or "Can this set be equipped with any smooth structure?". – Ennar Aug 18 '16 at 12:19

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