  Using the tractor calculus to study smooth metric measure spaces, we adapt
results of Gover and Nurowski to give sharp metric obstructions to the
existence of quasi-Einstein metrics on suitably generic manifolds. We do this
by introducing an analogue of the Weyl tractor $W$ to the setting of smooth
metric measure spaces. The obstructions we obtain can be realized as tensorial
invariants which are polynomial in the Riemann curvature tensor and its
divergence. By taking suitable limits of their tensorial forms, we then find
obstructions to the existence of static potentials, generalizing to higher
dimensions a result of Bartnik and Tod, and to the existence of potentials for
gradient Ricci solitons.
