  We employ the Bogoliubov approximation to study how the quantum geometry of
the helicity states affects the superfluid properties of a spin-orbit-coupled
Bose gas in continuum. In particular we derive the low-energy Bogoliubov
spectrum for a plane-wave condensate in the lower helicity band and show that
the geometric contributions to the sound velocity are distinguished by their
linear dependences on the interaction strength, i.e., they are in sharp
contrast to the conventional contribution which has a square-root dependence.
We also discuss the roton instability of the plane-wave condensate against the
stripe phase and determine their phase transition boundary. In addition we
derive the superfluid density tensor by imposing a phase-twist on the
condensate order parameter and study the relative importance of its
contribution from the interband processes that is related to the quantum
geometry.
