  We study multi-field inflation models that contain a non-trivial field-space
metric and a non-minimal coupling between the gravity and inflaton sectors. In
such models it is known that even in the absence of explicit interaction terms
the inflaton sector can decay into matter as a result of its non-minimal
coupling to gravity, thereby reheating the Universe gravitationally. Using the
Bogoliubov approach we evaluate the gravitational decay rates of the inflaton
fields into both scalars and fermions, and analyse the reheating dynamics. We
also discuss how the interpretation of the reheating dynamics differs in the
so-called Jordan and Einstein frames, highlighting that the calculation of the
Bogoliubov coefficients is independent of the frame in which one starts.
