locally conformal hyper-Kähler structures
New hyper-Kähler structures on tangent bundles
It is known that the tangent bundle \(TM\) of a Riemannian manifold \((M,g)\) is naturally endowed with an almost Kähler structure. However, such an almost Kähler structure is very rigid, since it is Kähler or it has constant scalar curvature only in the case when \((M,g)\) is flat. Thus, many authors have considered the problem of defining other almost Hermitian structures on \(TM\). In particular, in [\textit{M. Tahara} et al., Note Mat. 18, No. 1, 131--141 (1998; Zbl 0964.53021)] a class of almost hyper-Hermitian structures \((G,J_1,J_2,J_3)\) on \(TM\) was introduced.NEWLINENEWLINEIn this paper, some conditions ensuring that \((G,J_1,J_2,J_3)\) is locally conformal hyper-Kähler are given. As an application, a class of new hyper-Kähler structures is obtained on the tangent bundle of a complex space form (including the case of non-positive holomorphic sectional curvature).NEWLINENEWLINEFurther, using the above almost hyper-Hermitian structures \((G,J_1,J_2,J_3)\), a new class of almost contact metric \(3\)-structures \(\{(\phi_\alpha,\xi_\alpha,\eta_\alpha,\hat{G})\}_{\alpha\in\{1,2,3\}}\) on the unit tangent sphere bundle \(T_{1}M\) is discovered and, among them, a family of Sasakian \(3\)-structures on the unit tangent sphere bundle of a complex space form of positive holomorphic sectional curvature.
scientific article; zbMATH DE number 6359720
scientific article