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

E. S. Kornev

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 4 (2007), pp. 304-344

Résumé

In this work the special classes of left-invariant almost complex structures on four-dimensional Lie groups which are a direct product of smaller dimension groups are researched. Alongside with known classes of almost complex structures that hold the set left-invariant metric or the symplectic form, two new classes of almost complex structures called reduced and anti-reduced are defined. These structures are determined only by the representation of pair even-dimentional distributions of tangent subspaces. For reduced almost complex structures the exterior $2$-forms invariant concerning these structures and the associated left-invariant metrics are under construction. For each type of four-dimensional direct products of Lie groups the problem on an integrability of all classes of almost complex structures considered in work is researched, and there is an aspect of integrable structures. For the associated metrics the scalar curvature, sectional curvature, a Ricci tensor are calculated, and their properties are investigated. In work the classification theorem for orthogonal almost complex structures on four-dimensional Lie groups is proved, and also the theorem about an integrability of reduced almost complex structure on a Lie group having a event-dimentonal center is proved. Also are deduced the formulas expressing Neenhase tensor, scalar and sectional curvatures, a curvature and Ricci tensor through structural constants of a Lie algebra of a Lie group. Except for that the problem on existence of left-invariant symplectic structures, left-invariant Kahler, locally conformally Kahler and Einstein metrics are investigated.