Associated Nijenhuis Tensors on Manifolds with Almost Hypercomplex Structures and Metrics of Hermitian-Norden Type
arXiv:1510.00821 · doi:10.1007/s00025-016-0624-x
Abstract
An associated Nijenhuis tensor of endomorphisms in the tangent bundle is introduced. Special attention is paid to such tensors for an almost hypercomplex structure and the metric of Hermitian-Norden type. There are studied relations between the six associated Nijenhuis tensors as well as their vanishing. It is given a geometric interpretation of the vanishing of these tensors as a necessary and sufficient condition for the existence of a unique connection with totally skew-symmetric torsion tensor. Similar idea is used in the paper of T. Friedrich and S. Ivanov in Asian J. Math. (2002) for some other structures. Finally, an example of a 4-dimensional manifold of the considered type with vanishing associated Nijenhuis tensors is given.
13 pages