The Non-compact Case
A closely related question is the so-called "non-compact Yamabe problem", which asks: on a smooth, complete Riemannian manifold (M,g) which is not compact, does there exist a conformal metric of constant scalar curvature that is also complete? The answer is no, due to counterexamples given by Jin (1988).
Read more about this topic: Yamabe Problem
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“If there is a case for mental events and mental states, it must be that the positing of them, like the positing of molecules, has some indirect systematic efficacy in the development of theory.”
—Willard Van Orman Quine (b. 1908)
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