An analytical criterion for the completeness of Riemannian manifolds

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.