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We establish a weighted positive mass theorem which unifies and generalizesresults of Baldauf--Ozuch and Chu--Zhu. Our result is in fact equivalent to theusual positive mass theorem, and can be regarded as a positive mass theorem forsmooth metric measure spaces. We also study Dirac operators on certain warpedproduct manifolds associated to smooth metric measure spaces. Applications ofthis include, among others, an alternative proof for a special case of ourpositive mass theorem, eigenvalue bounds for the Dirac operator on closed spinmanifolds, and a new way to understand the weighted Dirac operator using warpedproducts.

Positive mass and Dirac operators on weighted manifolds and smooth metric measure spaces