General natural almost Hermitian and anti-Hermitian structures on the tangent bundles (Q2734141)

The author constructs some very general almost complex structures and semi-Riemannian metrics on the tangent bundle of a Riemannian manifold. The construction uses the theory of natural lifts and \(M\)-tensors [cf.\ \textit{K. P. Mok, E. M. Patterson} and \textit{Y. C. Wong}, Trans. Am. Math. Soc. 234, 253-278 (1977; Zbl 0363.53016)]. The almost complex structure (resp.\ the metric) depends on a set of four (resp.\ six) real valued, smooth parameters depending on a positive variable. The conditions satisfied by these parameters are then determined for the constructed structure to be (almost) Hermitian or Kählerian. In particular, some earlier work of the author [Public. Math. 55, 261-281 (1999; Zbl 0992.53053)] and \textit{M.~Tahara and Y.~Watanabe} [Math. J. Toyama Univ. 20, 149-60 (1997; Zbl 1076.53516)] are generalized. Finally, in a similar way, the author defines some anti-Hermitian and anti-Kählerian structures on the tangent bundle of a space form.