Invariants of Almost Complex and Almost Kahler Manifolds

Given a compact almost complex manifold Formula: see text, the almost complex invariant Formula: see text is defined as the complex dimension of the cohomology space Formula: see text. Its properties have been studied mainly when Formula: see text. If we endow Formula: see text with an almost Hermitian metric Formula: see text, then the number Formula: see text, i.e. the complex dimension of the space of Hodge–de Rham harmonic Formula: see text-forms, does not depend on the choice of almost Kähler metrics when Formula: see text. In this paper, we study the relationship between Formula: see text and Formula: see text in dimension Formula: see text. We prove Formula: see text if Formula: see text is non-integrable and observe that Formula: see text if the metric is almost Kähler. If Formula: see text is a compact quotient of a completely solvable Lie group and Formula: see text is a left-invariant almost Kähler structure on Formula: see text, we prove Formula: see text. Finally, we study the Formula: see text-pure and Formula: see text-full properties of Formula: see text on Formula: see text-forms for the special dimension Formula: see text.