  We present a general unified approach for the study of quantum thermal
machines, including both heat engines and refrigerators, operating under
periodic adiabatic driving and in contact with thermal reservoirs kept at
different temperatures. We show that many observables characterizing this
operating mode and the performance of the machine are of geometric nature.
Heat-work conversion mechanisms and dissipation of energy can be described,
respectively, by the antisymmetric and symmetric components of a thermal
geometric tensor defined in the space of time-dependent parameters generalized
to include the temperature bias. The antisymmetric component can be identified
as a Berry curvature, while the symmetric component defines the metric of the
manifold. We show that the operation of adiabatic thermal machines, and
consequently also their efficiency, are intimately related to these geometric
aspects. We illustrate these ideas by discussing two specific cases: a slowly
driven qubit asymmetrically coupled to two bosonic reservoirs kept at different
temperatures, and a quantum dot driven by a rotating magnetic field and
strongly coupled to electron reservoirs with different polarizations. Both
examples are already amenable for an experimental verification.
