A Connection With Parallel Totally Skew-Symmetric Torsion on a Class of Almost Hypercomplex Manifolds With Hermitian and Anti-Hermitian Metrics

Abstract

The subject of investigations are the almost hypercomplex manifolds with Hermitian and anti-Hermitian (Norden) metrics. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D and its torsion is totally skewsymmetric. The class of the nearly Kähler manifolds with respect to the first almost complex structure is of special interest. It is proved that D has a D-parallel torsion and is weak if it is not flat. Some curvature properties of these manifolds are studied. Int. J. Geom. Methods Mod. Phys. 2011.08:115-131. Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY -SERIALS UNIT on 03/11/15. For personal use only. The almost (H, G)-manifoldsLet (M, H) be an almost hypercomplex manifold, i.e. M is a 4n-dimensional differentiable manifold and H = (J 1 , J 2 , J 3 ) is a triple of almost complex structures with the properties:for all cyclic permutations (α, β, γ) of (1, 2, 3) and I denotes the identity ([6, 1]). The standard structure of H on a 4n-dimensional vector space with a basis {X 4k+1 , X 4k+2 , X 4k+3 , X 4k+4 } k=0,1,...,n−1 has the form [37]:Further, x, y, z, w will stand for arbitrary differentiable vector fields on M . Let g be a pseudo-Riemannian metric on (M, H) with the properties

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

“…Then, taking into account (5) and (20), we obtain Proof. The statement follows from Proposition 3 and Proposition 4, bearing in mind (18) and (20). Proof.…”

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

“…Then, we call that the almost hypercomplex manifold is equipped with an HN-metric structure (HN is an abbreviation for Hermitian-Norden). Namely, the metric g is Hermitian for α = 1, whereas g is a Norden metric in the cases α = 2 and α = 3 ([8], [17]). Let us consider (M, J 1 , g) in the class G 1 (the so-called cocalibrated manifolds with Hermitian metric), according to the classification in [7], as well as (M, J α , g), α = 2; 3, in the class W 3 (the so-called quasi Kähler manifolds with Norden metric), according to the classification in [5].…”

“…The geometry of a metric connection with totally skew-symmetric torsion preserving the almost hypercomplex structure with Hermitian metric is introduced in [11]. A connection of such type, considered on almost hypercomplex manifolds with Hermitian and Norden metrics, is investigated in [17].…”

On the geometry of connections with totally skew-symmetric torsion on manifolds with additional tensor structures and indefinite metrics

“…We have proved the following In [15] we have constructed a connection D, using the KT-connections D 1 , D 2 and D 3 , on an almost (H, G)-manifold from the class W 133 and we have proved the following Theorem 17 ([15]). The connection D defined by g (D x y, z) = g (∇ x y, z) + 1 2 F 1 (x, y, J 1 z).…”