Remarks on some almost Hermitian structure on the tangent bundle. II (Q466167)

Let \(M\) and \(TM\) be an almost Hermitian manifold and its tangent bundle respectively. \textit{M. Tahara} et al. [Note Mat. 18, No. 1, 131--141 (1998; Zbl 0964.53021)] constructed three almost complex structures \(J_1,J_2,J_3\) on \(TM\), which define in special cases an almost hyper-complex structure. They also found a Riemannian metric \(G_1\) such that \((J_1,G_1)\) is an almost Hermitian structure on \(TM\). NEWLINENEWLINENEWLINEHere, the authors construct Riemannian metrics \(G_2\) and \(G_3\) on \(TM\) such that \((J_2,G_2)\) and \((J_3,G_3)\) are almost Hermitian structures on \(TM\). For these structures \((J_i,G_i)\), \(i=1,2,3\), they find conditions such that \((TM,J_i,G_i)\) belongs to the classes constructed by \textit{A. Gray} and \textit{L. M. Hervella} [Ann. Mat. Pura Appl. (4) 123, 35--58 (1980; Zbl 0444.53032)].