Synectic metrics in tangent bundle of almost Hermitian space of hyperbolic type (Q677766)

The author considers for an even-dimensional Riemannian manifold \(V_n\) with metric tensor \(g_{ij}\) and with almost double structure \(f^i_j(f^s_jf^i_s= \delta^i_j\), \(f^s_s=0)\) in the tangent bundle \(T(V_n)\) the ``synectic metric'' \({\mathcal G}={^cg}+{^va}\) and the almost product structure \({\mathcal F}={^cf}+ {^vA}\), where \({^cg}\), \({^cf}\) are the complete lifts in \(T(V_n)\) of the metric tensor \(g\) and of the structure affinor \(f\), and \({^va}\) and \({^vA}\) are the vertical lifts in \(T(V_n)\) of a certain symmetric tensor \(a\) and a special affinor \(A\) on the base. The author studies the conditions for different types of almost Hermitian metrics \({\mathcal G}\) in \(T(V_n)\).