  We report on a lattice fermion formulation with a curved domain-wall mass
term to nonperturbatively describe fermions in a gravitational background. In
our previous work in 2022, we showed under the time-reversal symmetry that the
edge-localized massless Dirac fermion appears on one and two-dimensional
spherical domain-walls and the spin connection is induced on the lattice in a
consistent way with continuum theory. In this work, we extend our study to the
Shamir type curved domain-wall fermions without the time-reversal symmetry. We
find in the free fermion case that a single Weyl fermion appears on the edge,
and feels gravity through the induced spin connection. With a topologically
nontrivial $U(1)$ gauge potential, however, we find an oppositely chiral zero
mode at the center where the gauge field is singular.
