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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?

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