# Completeness of the tangent bundle of riemannian manifold

Let $$(M,g)$$ be a riemannian manifold and $$TM$$ its tangent bundle. There are natural riemannian metrics that we can endow the tangent bundle with (for instance the Sasaki metric) and I wonder if for some of them $$TM$$ is complete (under maybe some suitable additional assumptions about $$M$$). This means that $$TM$$ is complete as a metric space, which is equivalent with geodesic completeness (by Hopf-Rinow theorem). Does anyone have an idea or reference?