A flat two-torus, $T^2$ that is the torus with Euclidean metric needs to be embedded at least 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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2It is a bit strange to call the flat metric on the torus the Euclidean metric... – Mariano Suárez-Álvarez Feb 14 '11 at 06:01
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Thinking of $T^2$ as $\mathbb R^2/\mathbb Z^2$, the term Euclidean metric makes some sense. :) – Cheerful Parsnip Feb 14 '11 at 11:23