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.