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?
Embedding of punctured torus with euclidean metric
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riemannian-geometry
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2It is a bit strange to call the flat metric on the torus the *Euclidean metric*... – 2011-02-14
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0Thinking of $T^2$ as $\mathbb R^2/\mathbb Z^2$, the term Euclidean metric makes some sense. :) – 2011-02-14