The main purpose of the present paper is to study the geometry of Riemannian manifolds endowed with Golden structures. We discuss the problem of integrability for Golden Riemannian structures by using a f-operator which is applied to pure tensor fields. Also, the curvature properties for Golden Riemannian metrics and some properties of twin Golden Riemannian metrics are investigated. Finally, some examples are presented.

https://journals.tubitak.gov.tr/math/issues/mat-13-37-4/mat-37-4-14-1108-35.pdf

On integrability of Golden Riemannian structures

This paper is concerned with the problem of the geometry of Norden manifolds. Some properties of Riemannian curvature tensors and curvature scalars of Kahler-Norden manifolds using the theory of Tachibana operators is presented.

On Kähler-Norden manifolds

On the basis of the phase completion the notion of vertical and horizontal lifts of vector fields is defined in the tensor bundles over a Riemannian manifold. Such a tensor bundle is made into a manifold with a Riemannian structure of special type by endowing it with Sasakian metric. The components of the Levi-Civita and other metric connections with respect to Sasakian metrics on tensor bundles with respect to the adapted frame are presented. This having been done, it is shown that it is possible to study geodesics of Sasakian metrics dealing with geodesics of the base manifolds.

Geodesics of Sasakian Metrics on Tensor Bundles

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.

Notes on para-norden-walker 4-manifolds ∗

Curvature properties are studied for the Sasaki metric on the (1, 1) tensor bundle of a Riemannian manifold. As an application, examples of almost para-Nordenian and para-Kahler-Nordenian B-metrics are constructed on the (1, 1) tensor bundle by looking at the Sasaki metric. Also, with respect to the para-Nordenian B-structure, paraholomorphic conditions for the complete lifts of vector fields are analyzed.

On the geometry of the (1, 1)-tensor bundle with Sasaki type metric

Paraholomorphic B-manifold and its properties ✩

The main purpose of the present paper is to study almost Norden structures on 4-dimensional Walker manifolds. We discuss the integrability and Kahler (holomorphic) conditions for these structures. The curvature properties for Norden-Walker metrics is also investigated. Then examples Norden-Walker metrics are constructed from an arbitrary harmonic function of two variables.

/pdf/some-properties-of-norden-walker-metrics-3ucjchonvb.pdf

Some properties of Norden-Walker metrics

The main purpose of this article is to investigate the paraholomorphy property of the Sasaki and Cheeger–Gromoll metrics by using compatible paracomplex stuctures on the tangent bundle.

/pdf/on-para-kahler-norden-structures-on-the-tangent-bundles-362jomg2gl.pdf

Murat Iscan

Papers

Paraholomorphic B-manifold and its properties ✩

On Kähler-Norden manifolds

Notes on para-norden-walker 4-manifolds ∗

Some properties of Norden-Walker metrics

On para-Kähler-Norden structures on the tangent bundles