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An analytical criterion for the completeness of Riemannian manifolds


Author: William B. Gordon
Journal: Proc. Amer. Math. Soc. 37 (1973), 221-225
MSC: Primary 53C20
DOI: https://doi.org/10.1090/S0002-9939-1973-0307112-5
MathSciNet review: 0307112
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Abstract: If M is a (not necessarily complete) riemannian manifold with metric tensor $ {g_{ij}}$ and f is any proper real valued function on M, then M is necessarily complete with respect to the metric $ {\tilde g_{ij}} = {g_{ij}} + (\partial f/\partial {x^i})(\partial f/\partial {x^j})$. Using this construction one can easily prove that a riemannian manifold is complete if and only if it supports a proper function whose gradient is bounded in modulus.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0307112-5
Article copyright: © Copyright 1973 American Mathematical Society

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