An analytical criterion for the completeness of Riemannian manifolds

Abstract

If M is a (not necessarily complete) riemannian manifold with metric tensor g i j {g_{ij}} and f is any proper real valued function on M, then M is necessarily complete with respect to the metric g ~ i j = g i j + ( ∂ f / ∂ x i ) ( ∂ f / ∂ x j ) {\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.