Associated Nijenhuis Tensors on Manifolds with Almost Hypercomplex Structures and Metrics of Hermitian–Norden Type

Abstract

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.

Manifolds with almost contact 3-structure and metrics of Hermitian–Norden type

“…In [16], it is proven the following Proposition 2. Let (J 1 , J 2 , J 3 ) be an almost hypercomplex structure and G is a pseudo-Riemannian metric on the almost hypercomplex manifold.…”

“…Since any {J α , J α } is a tensor field of type (1,2), it suffices to compute the tensors {J α , J α }((x, 0), (y, 0)) and {J α , J α }((x, 0), (o, d dt )), where o is the zero element of X(M). Taking into account (10), (15), (16), (17), we obtain consequently: Proof. We set y = ξ α in N (1) α (x, y) = 0 and apply η α .…”

“…According to [7], the G 1 -manifolds are almost Hermitian manifolds whose corresponding Nijenhuis (0,3)-tensor by the Hermitian metric is a 3-form. This condition is equivalent to the vanishing of the associated Nijenhuis tensor, according to [16].…”

Almost Complex and Hypercomplex Norden Structures Induced by Natural Riemann Extensions

“…Differentiable manifolds M 4n equipped with structures (H, G) = (J 1 , J 2 , J 3 , g, Φ, g 2 , g 3 ) are studied in Refs. [23][24][25][26][27][28] under the name almost hypercomplex pseudo-Hermitian manifolds, almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics, and almost hypercomplex manifolds with Hermitian-Norden metrics, respectively. In this paper we refer to (M 4n , H, G) as an almost hypercomplex manifold with Hermitian-Norden metrics.…”