Abstract: A solvmanifold is one of 4-dimensional geometries
in the sense of Thurston, which do not admit a complex structure compatible with
group of isometries. It is shown that an almost neutral Hermitian structure
chosen naturally on is neither neutral Hermitian nor
almost neutral Kähler, and similarly that an opposite almost neutral Hermitian
structure chosen naturally on is neither opposite neutral
Hermitian nor opposite almost neutral Kähler. Moreover, two kinds of almost
neutral Kähler structures are not isotropic Kähler.
Keywords and phrases: solvable Lie group, solvmanifold, geometric structures on 4-manifolds, almost neutral Kähler structure, opposite almost neutral Kähler structure.