An Analytical Necessary and Sufficient Condition for a Riemannian Manifold to be Complete.
NAVAL RESEARCH LAB WASHINGTON D C MATHEMATICS RESEARCH CENTER
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It is known that every differentiable manifold supports a complete Riemannian structure. This is a consequency of Whitneys Embedding Theorem. Moreover, Nomizu and Ozeki have shown that every Riemannian manifold is conformally equivalent to a complete Riemannian manifold. In this report a method is given for constructing complete Riemannian metrics which is exceedingly simple and provides a necessary and sufficient condition for the completeness of a Riemannian structure. Namely, it is phonon that a Riemannian manifold is complete if and only if it supports a proper function whose gradient is bounded in modulus.
- Theoretical Mathematics