The existence of complete Riemannian metric

The answer is yes, altought the construction is not simple. Given any metric on a differential manifold, it can be shown that it is conformally equivalent to a complete riemannian manifold.

Reference: Nomizu, Katsumi, and Hideki Ozeki. "The existence of complete Riemannian metrics." Proceedings of the American Mathematical Society 12.6 (1961): 889-891.

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