  In this thesis, we consider fermion systems on square lattice spaces with a
curved domain-wall mass term. In a similar way to the flat case, we find
massless and chiral states localized at the wall. In the case of $S^1$ and
$S^2$ domain-wall embedded into a square lattice, we find that these edge
states feel gravity through the induced spin connection. In the conventional
continuum limit of the higher dimensional lattice, we find a good consistency
with the analytic results in the continuum theory. We also confirm that the
rotational symmetry is recovered automatically.
  We also discuss the effect of a $U(1)$ gauge connection on a two-dimensional
lattice fermion with the $S^1$ domain-wall mass term. We find that the gauge
field changes the eigenvalue spectrum of the boundary system by the
Aharanov-Bohm effect and generates an anomaly of the time-reversal ($T$)
symmetry. Our numerical evaluation is consistent with the Atiyah-Patodi-Singer
index, which describes the cancellation of the $T$ anomaly by the topological
term on the bulk system. When we squeeze the flux inside one plaquette while
keeping the total flux unchanged, the anomaly inflow undergoes a drastic
change. The intense flux gives rise to an additional domain wall around the
flux. We observe a novel localized mode at the flux, canceling the $T$ anomaly
on the wall instead of the topological term in the bulk.
  We apply the study to a problem in condensed matter physics. It is known that
inside topological insulators, a vortex or monopole acquires a fractional
electric charge and turns into a dyon. Describing the topological insulator as
a negative mass region of a Dirac fermion, we provide a microscopic description
of this phenomenon in terms of the dynamical domain-wall creation.
