  We consider a weakly-interacting Bose-Einstein condensate (BEC) that is
loaded into an optical lattice with a two-point basis, and described by a
two-band Bose-Hubbard model with generic one-body and two-body terms. By first
projecting the system to the lower Bloch band and then applying the Bogoliubov
approximation to the resultant Hamiltonian, we show that the inverse
effective-mass tensor of the superfluid carriers in the Bogoliubov spectrum has
two physically distinct contributions. In addition to the usual inverse
band-mass tensor that is originating from the intraband processes within the
lower Bloch band, there is also a quantum-geometric contribution that is
induced by the two-body interactions through the interband processes. We also
discuss the conditions under which the latter contribution is expressed in
terms of the quantum-metric tensor of the Bloch states, i.e., the natural
Fubini-Study metric on the Bloch sphere.
