Invariant characteristics of some classes of almost Hermitian structures

On the tangent bundle $TM$ of a smooth manifold $M$ with a nonlinear connection $\nabla$ and a generalized Lagrangian metric $g$ we consider a Riemannian metric $\tilde g$ such that
$$ \tilde g(X^h,Y^h)=\tilde g(X^v,Y^v)=g(X,Y), \qquad \tilde g(X^h,Y^v)=0, $$
where $X^h,Y^h$ and $X^v,Y^v$ are, respectively, the horizontal and vertical lifts of vector fields $X$ and $Y$ on $M$. The metric $\tilde g$ is Hermitian with respect to the almost complex structure $J$: $JX^h=X^v$, $JX^v=-X^h$. We find invariant characteristics of certain classes of almost Hermitian structures $(TM,\tilde g,J)$, e.g. Kahlerian structures, almost Kahlerian structures, semi-Kahlerian structures.