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Math question: are riemannian geometries necessarily finite and bounded like a sphere? Or can you have infinite ones as with Euclidean or hyperbolic?
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The terms "Euclidean" and "hyperbolic" are downstream of a riemannian metric. The curvature of the riemannian metric being 0 or negative defines Euclidean or hyperbolic respectively. So yes you can have non compact Riemannian geometries. 1/
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Is it possible to have non compact (compact is I am guessing what you probably meant by "finite and bounded") positive curvature riemannian metrics? Yes. take a sphere, remove a point from it. This turns out to be positively curved but is not "compact"
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Remove two points - the north and south pole - from a sphere. Now imagine that every time you circumnagivate the equator you go down or up one level (i.e. like an onion with infinitely many layers). This is the universal cover of S^2-{N,S}
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I don't know the answer off the top of my head (for the general case) but I suspect some smart application of the generalized Gauss Bonnet theorem might yield an answer. (Intuitively, my hunch is yes but I could be wildly wrong on this)