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Use this for questions pertaining to curvature of manifolds. Does not need to be specific to general relativity, but also for curvature of e.g. a Calabi-Yau manifold.

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Are there closed simply connected 2D manifolds that do not require a third dimension?

The other aspect is to do with the metric definition of curvature. … , and longer than $2\pi r$ with negative curvature. …
Nullius in Verba's user avatar