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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8
votes
Two discs with no parallel tangent planes
If I understood your question correctly, I think that the answer is no. In fact, even more is true: if you choose any identification $\varphi \colon \Sigma_1 \to \Sigma_2$ in such a way that the boun …
19
votes
Compactification of a manifold
A "surface of infinite genus" $S$ is an example of a manifold that is not an open subset of a compact manifold.
The reason $S$ cannot be embedded in a compact manifold is straightforward: we can find …