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This is a closely related question:

Existence of closed surface having only negative gaussian curvature.

The answer is that such surface ($2$-D manifold) does not exist if we require it to be embedded to $\mathbb R^3$.

However I am still wondering whether there are closed $2$-D manifolds embedded in $\mathbb R^3$ with only countable number of points where the curvature is nonnegative and if it doesn't exist, why?

I am specializing in theoretical chemistry and am a math hobbyist so please feel free to point it out if there is any misunderstanding or any blatant error. Thanks in advance.

Weijun Zhou
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    By closed surface do you mean a (smooth) compact surface without boundary? If so, it must have uncountably many points of positive curvature. – Ted Shifrin Jan 23 '18 at 01:52
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    Once you know the curvature is positive at a point, continuity of the curvature tells you it's positive in a small disc around that point, which contains an uncountable number of points. – Anthony Carapetis Jan 23 '18 at 01:53
  • @AnthonyCarapetis I see. Sorry I have failed to see that. Thank you very much for enlightening me. – Weijun Zhou Jan 23 '18 at 02:21

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