  We present a study on physical properties for a large distant red galaxy
(DRG) sample, using the $K$-selected multi-band photometry catalog of the
COSMOS/UltraVISTA field and the CANDELS NIR data. Our sample includes 4485 DRGs
with $(J-K)_\mathrm{AB}>1.16$ and $K_\mathrm{AB}<$23.4 mag, and 132 DRGs have
HST/WFC3 morphological measurements. The results of nonparametric measurements
of DRG morphology are consistent with our rest-frame UVJ color classification:
quiescent DRGs are generally compact while star-forming DRGs tend to have
extended structures. We find the star formation rate (SFR) and the stellar mass
of star-forming DRGs present tight "main sequence" relations in all redshift
bins. Moreover, the specific SFR (sSFR) of DRGs increase with redshift in all
stellar mass bins and DRGs with higher stellar masses generally have lower
sSFRs, which indicates that galaxies were much more active on average in the
past, and star formation contributes more to the mass growth of low-mass
galaxies than to high-mass galaxies. The infrared (IR) derived SFR dominate the
total SFR of DRGs which occupy the high-mass range, implying that the $J-K$
color criterion effectively selects massive and dusty galaxies. DRGs with
higher $M_{*}$ generally have redder $(U-V)_\mathrm{rest}$ colors, and the
$(U-V)_\mathrm{rest}$ colors of DRGs become bluer at higher redshifts,
suggesting high-mass galaxies have higher internal dust extinctions or older
stellar ages and they evolve with time. Finally, we find that DRGs have
different overlaps with EROs, BzKs, IEROs and high-$z$ ULIRGs indicating DRGs
is not a special population and they can also be selected by other color
criteria.
