  We report a study describing the charm quark by a domain-wall fermion (DWF)
in lattice quantum chromodynamics (QCD). Our study uses a quenched gauge
ensemble with the DBW2 rectangle-improved gauge action at a lattice cutoff of
$a^{-1} \sim 3$ GeV. We calculate masses of heavy-light (charmed) and
heavy-heavy (charmonium) mesons with spin-parity $J^P = 0^\mp$ and $1^\mp$,
leptonic decay constants of the charmed pseudoscalar mesons ($D$ and $D_s$),
and the $D^0$-$\bar{D^0}$ mixing parameter. The charm quark mass is found to be
$m^{\bar{\rm MS}}_{c}(m_{c})=1.24(1)(18)$ GeV. The mass splittings in
charmed-meson parity partners $\Delta_{q,J=0}$ and $\Delta_{q, J=1}$ are
degenerate within statistical errors, in accord with experiment, and they
satisfy a relation $\Delta_{q=ud, J} > \Delta_{q=s, J}$, also consistent with
experiment. A C-odd axial vector charmonium state, $h_c), lies 22(11) MeV above
the $\chi_{c1}$ meson, or $m_{h_{c}} = 3533(11)_{\rm stat.}$ MeV using the
experimental $\chi_{c1}) mass. However, in this regard, we emphasize
significant discrepancies in the calculation of hyperfine splittings on the
lattice. The leptonic decay constants of $D$ and $D_s$ mesons are found to be
$f_D=232(7)_{\rm stat.}(^{+6}_{-0})_{\rm chiral}(11)_{\rm syst.}$ MeV and
$f_{D_s}/f_{D} = 1.05(2)_{\rm stat.}(^{+0}_{-2})_{\rm chiral}(2)_{\rm syst.}$,
where the first error is statistical, the second a systematic due to chiral
extrapolation and the third error combination of other known systematics. The
$D^0$-$\bar{D^0}$ mixing bag parameter, which enters the $\Delta C = 2$
transition amplitude, is found to be $B_D(2{GeV})=0.845(24)_{\rm
stat.}(^{+24}_{-6})_{\rm chiral}(105)_{\rm syst.}$.
