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A flat two-torus, $T^2$ that is the torus with Euclidean metric needs to at least be embedded in $\mathbb{R}^4$. If we puncture the torus and leave the Euclidean metric on it as inherited (ignoring the issues of completeness), what ambient space could you embed the punctured torus with Euclidean metric?

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It is a bit strange to call the flat metric on the torus the Euclidean metric... – Mariano Suárez-Alvarez Feb 14 '11 at 6:01
Thinking of $T^2$ as $\mathbb R^2/\mathbb Z^2$, the term Euclidean metric makes some sense. :) – Grumpy Parsnip Feb 14 '11 at 11:23

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