  Recent experiment on a honeycomb-lattice Heisenberg antiferromagnet (AF)
Bi$_3$Mn$_4$O$_{12}$(NO$_3$) revealed a novel spin-liquid-like behavior down to
low temperature, which was ascribed to the frustration effect due to the
competition between the AF nearest- and next-nearest-neighbor interactions
$J_1$ and $J_2$. Motivated by the experiment, we study the ordering of the
$J_1$ -$J_2$ frustrated classical Heisenberg AF on a honeycomb lattice both by
a low-temperature expansion and a Monte Carlo simulation. The model has been
known to possess a massive degeneracy of the ground state, which, however,
might be lifted due to thermal fluctuations leading to a unique ordered state,
the effect known as 'order-by-disorder'. We find that the model exhibits an
intriguing ordering behavior, particularly near the AF phase boundary. The
energy scale of the order-by-disorder is suppressed there down to extremely low
temperatures, giving rise to exotic spin-liquid states like a "ring-liquid" or
a "pancake-liquid" state accompanied by the characteristic spin structure
factor and the field-induced antiferromagnetism. We argue that the recent
experimental data are explicable if the system is in such exotic spin-liquid
states.
