Example: take curved 2D space with positive constant curvature everywhere. You say, sphere with radius R? no, there are(adsbygoogle = window.adsbygoogle || []).push({}); 2 differentsolutions in topology: sphere and half-sphere. Half sphere (1/2 of sphere where points across the 'equator' are connected to the opposite sides) can’t be 'embedded' in 3D 'continuously'. Both objects have different extrinsic properties and total volume (so the difference can be discovered by an observer ‘inside’) but the same curvature everywhere.

I’ve heard that for 3D, and especially hyperbolic 4D spaces (like ours) it is much worse – there are infinitely many different solutions with different extrinsic properties.

So, my question is – is anyone working on it? Any links, articles? I was always wondering… say, 2 sides of curved spacetime intersect in the embedded higher dimensional space. Does it mean that these 2 points meet in our physical spacetime? Or (if embedding is a pure abstraction) they can go thru each other without any interaction (like in the Klein bottle)

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# Extrinsic properties of the curved space

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