  In this paper, we consider the Boltzmann equation with respect to orthonormal
vielbein fields in conservative form. This formalism allows the use of
arbitrary coordinate systems to describe the space geometry, as well as of an
adapted coordinate system in the momentum space, which is linked to the
physical space through the use of vielbeins. Taking advantage of the
conservative form, we derive the macroscopic equations in a covariant tensor
notation, and show that the hydrodynamic limit can be obtained via the
Chapman-Enskog expansion in the Bhatnaghar-Gross-Krook (BGK) approximation for
the collision term. We highlight that in this formalism, the component of the
momentum which is perpendicular to some curved boundary can be isolated as a
separate momentum coordinate, for which the half-range Gauss-Hermite quadrature
can be applied. We illustrate the capabilities of this formalism by considering
two applications. The first one is the circular Couette flow between rotating
coaxial cylinders, for which benchmarking data is available for all degrees of
rarefaction, from the hydrodynamic to the ballistic regime. The second
application concerns the flow in a gradually expanding channel. We employ
finite-difference lattice Boltzmann models based on half-range Gauss-Hermite
quadratures for the implementation of diffuse reflection, together with the
fifth order WENO and third-order TVD Runge-Kutta numerical methods for the
advection and time-stepping, respectively.
