Pseudo-Chern classes of an almost pseudo-Hermitian manifold

For an almost pseudo-Hermitian manifold, pseudo-Chern classes are defined on its complexified tangent bundle with the pseudo-Hermitian structure as represented by certain ${\text{ad}}(U(p,\,q))$-invariant forms on the manifold. It is shown that such a manifold always admits an almost Hermitian structure, and hence that Chern classes are also defined on the complexified tangent bundle with such an almost Hermitian structure. A relation between the pseudo-Chern classes and the Chern classes is established. From the relation, the pseudo-Chern classes are considered as the characteristic classes which measure how the almost pseudo-Hermitian structure deviates from an almost Hermitian structure.