  In this paper, we characterize Riemannian 4-manifold in terms of its almost
Hermitian twistor spaces $(Z,g_t,\mathbb{J}_{\pm})$. Some special metric
conditions (including Balanced metric condition, first Gauduchon metric
condition) on $(Z,g_t,\mathbb{J}_{\pm})$ are studied. For the first Chern form
of a natural unitary connection on the vertical tangent bundle over the twistor
space $Z$, we can recover J. Fine and D. Panov's result on the condition of the
first Chern form being symplectic and P. Gauduchon's result on the condition of
the first Chern form being a (1,1)-form respectively, by using the method of
moving frames.
