Left-invariant almost complex structures and the associated metrics on four-dimensional Lie groups

In the work left-invariant almost complex structures and the left-invariant metrics corrisponding to these structures on four-dimensional Lie groups are researched. First, classification of orthogonal concerning the left-invariant metric almost complex structures is given, depending on a signature of the metric in a case of dimension 4. Then two new classes of almost complex structures which are called reduced and anti-reduced are introduced. These almost complex structures are completely determined by pair the set two-dimensional distributions of tangents subspaces. In a case of dimension 4 there is an aspect of these structures in the fixed base and is shown that each such structure holds some left-invariant exterior 2-form. With reduced almost complex structures left-invariant metrics is associated. These metrics are interesting to those that with their help on Lie groups it is possible to obtain left-invariant Einstein, Kahler and locally conformally Kahler metrics. In the work formulas which Nijenhuis tensor of left-invariant almost complex structure and basic geometrical characteristics of the associated metrics (such as a Ricci tensor, a scalar curvature and sectional curvature) with structural constants of a Lie algebra of the Lie group also are deduced express. The theorem of an integrability of reduced almost complex structure on a Lie group with the non-trivial center is proved. In the work detailed examples of reduced and antireduced almost complex structures and the associated metrics are represented, in a case when the Lie group is a direct product, and in a case when the Lie group is semi-direct product.