Arif Salimov

NOTES ON PARA-NORDEN-WALKER 4-MANIFOLDS ∗

A Walker 4-manifold is a pseudo-Riemannian manifold, (M4, g) of neutral signature, which admits a field of parallel null 2-plane. The main purpose of the present paper is to study almost paracomplex structures on 4-dimensional Walker manifolds. We discuss sequently the problem of integrability, para-Kahler (paraholomorphic), quasi-para-Kahler and isotropic para-Kahler conditions for these structures. The curvature properties for para-Norden–Walker metrics with respect to the almost paracomplex structure and some properties of para-Norden–Walker metrics in context of almost product Riemannian manifolds are also investigated. Also, we discuss the Einstein conditions for these structures.

DIAGONAL LIFTS OF TENSOR FIELDS OF TYPE (1;1) ON CROSS-SECTIONS IN TENSOR BUNDLES AND ITS APPLICATIONS

The main purpose of this paper is to investigate diagonal lift of tensor fields of type (1, 1) from manifold to its tensor bundle of type (p, q) and to prove that when a manifold Mn admits a Kahlerian structure (φ, g), its tensor bundle of type (p, q) admits an complex structure.

A NOTE ON THE GOLDBERG CONJECTURE OF WALKER MANIFOLDS

This paper is concerned with Goldberg conjecture. Using the ϕφ-operator we prove the following result. Let (M, φ, w g) be an almost Kahler–Walker–Einstein compact manifold with the proper almost complex structure φ. The proper almost complex structure φ on Walker manifold (M, w g) is integrable if ϕφgN+ = 0, where gN+ is the induced Norden–Walker metric on M. This resolves a conjecture of Goldberg under the additional restriction on Norden–Walker metric (gN+ ∈ Ker ϕφ).

NON-EXISTENCE OF PARA-KAHLER–NORDEN WARPED METRICS

In this paper, we show that para-Kahler–Norden (or paraholomorphic) metrics do not exist on warped manifolds.

Geodesics for complete lifts of affine connections in tensor bundles

The purpose of the present paper is to define using the @c-operator [Bol. Soc. Mat. Mexic. 8 (3) (2002) 75] complete lifts of affine connections to the tensor bundle and investigate their geodesics.

SOME NOTES ON ALMOST PARACONTACT STRUCTURES

In the present paper we study an almost paracontact Riemannian manifold which is closely related to the pure Riemannian metric. We construct a new tensor field corresponding to pure metric of almost paracontact structure. An integrability condition and curvature properties of structure by using this tensor field are presented. Finally, we study almost paracontact structures with structural exact 1-form.

ON BICOMPLEX-HOLOMORPHIC MANIFOLDS

The main purpose of this paper is to study some properties of bicomplex-holomorphic 4-manifolds endowed with a holomorphic double Walker–Norden metric.

Horizontal lift of affinor structures and its applications

The main purpose of the present paper is to study the horizontal lifts of tensor field of type (1, 1) (affinor field) to tensor bundle and the integrability conditions for the horizontal lifts of special types of complex and tangent structures.

On structure-preserving connections

In this paper we find the formula of connections under which an almost complex structure is covariantly constant. These types of connections on anti-Kähler–Codazzi manifolds are described. Also, twin metric-preserving connections are analyzed for quasi-Kähler manifolds. Finally, anti-Hermitian Chern connections are investigated.