  We report a study of magnetism and magnetic transitions of hexagonal
ErMnO$_3$ single crystals by magnetization, specific heat and heat transport
measurements. Magnetization data show that the $c$-axis magnetic field induces
three magnetic transitions at 0.8, 12 and 28 T. The specific heat shows a peak
at 2.2 K, which is due to a magnetic transition of Er$^{3+}$ moments. For
low-$T$ thermal conductivity ($\kappa$), a clear dip-like feature appears in
$\kappa(H)$ isotherm at 1--1.25 T for $H \parallel ab$; while in the case of $H
\parallel c$, a step-like increase is observed at 0.5--0.8 T. The transition
fields in $\kappa(H)$ are in good agreement with those obtained from
magnetization, and the anomaly of $\kappa$ can be understood by a spin-phonon
scattering scenario. The natures of magnetic structures and corresponding
field-induced transitions at low temperatures are discussed.
