On conharmonic transformations of weyl spaces

It has been found that an almost complex structure of a contact metric manifold on frame bundle ( F M , g D , J ) is an almost Hermitian manifold. The derivative and coderivative of the Kahler form of the almost Hermitian structure ( g D , J ) are determined on frame bundle. An almost complex structure is a particular case of the polynomial structure of degree 2 satisfying J 2 = p J + q I , where p = 0 , q = − 1 . However, the main contribution of this paper is that the results by applying the p , q as positive numbers then it satisfies the condition on J 2 = p J + q I and termed as metallic structure. Furthermore, a tensor field J ˜ is introduced on a frame bundle F M which proves that it is metallic structure on F M . The proposed theorem shows that the diagonal lift g D of a Riemannian metric g is a metallic Riemannian metric on F M . The derivative and coderivative of 2-form F of metallic Riemannian structure on F M are calculated. Moreover, the Nijenhuis tensor of tensor field J ˜ is determined. Finally, a locally metallic Riemannian manifold ( F M , J H , g D ) is described as an application.

Novel theorems for the frame bundle endowed with metallic structures on an almost contact metric manifold

In this paper firstly, the Tachibana operators were applied t o 1−form, vertical, complete and horizontal lifts with respect to almost paracomplex structure I D (The diagonal lift I D ) on cotangent bundle. Secondly, the Vishnevskii operators were applied to 1−form according to the diagonal lift I D on cotangent bundle. Finally, covariant derivatives of almost paracomplex structure I D with respect to vertical, complete and horizontal lifts were obt ained.

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Some notes on almost paracomplex structures associated with the diagonal lifts and operators on

If (M,∇) is a manifold with a symmetric linear connection, then T*M can be endowed with the natural Riemann extension $$\bar g$$
 (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $$\bar g$$
 initiated by C. L.Bejan and O.Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $$\mathcal{P}$$
 on (T*M, $$\bar g$$
 
) and prove that $$\mathcal{P}$$
 is harmonic (in the sense of E.Garcia-Rio, L.Vanhecke and M. E.Vazquez-Abal (1997)) if and only if $$\bar g$$
 reduces to the classical Riemann extension introduced by E.M. Patterson and A.G. Walker (1952).

/pdf/a-characterization-of-the-riemann-extension-in-terms-of-2d76skdq64.pdf

A characterization of the Riemann extension in terms of harmonicity

To get exact solutions to Einstein’s field equations in general relativity, one has to impose some symmetry requirements. Otherwise, the equations are too difficult to solve. However, sometimes, the imposition of too much extra symmetry can cause the problem to become somewhat trivial. As a typical example to illustrate this, the effects of conharmonic flatness are studied and applied to Friedmann–Lemaitre–Robertson–Walker spacetime. Hence, we need to impose some symmetry to make the problem tractable, but not too much so as to make it too simple.

/pdf/how-extra-symmetries-affect-solutions-in-general-relativity-1md2j01swr.pdf

How Extra Symmetries Affect Solutions in General Relativity

In this paper, we consider a pseudo-Riemannian manifold equipped with a Kahler–Norden–Codazzi golden structure. For such a manifold, we study curvature properties. Also, we define special connections of the first type and of the second type on the manifold, which preserve the associated twin Norden golden metric and satisfy some special conditions and present some results concerning them.

On Kähler-Norden-Codazzi golden structures on pseudo-Riemannian manifolds

Let $M$ be an $n-$dimensional differentiable manifold with a symmetric connection $\nabla $ and $T^{\ast}M$ be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension $% \widetilde{g}_{\nabla,c}$ on $T^{\ast}M$ defined by means of a symmetric $% (0,2)$-tensor field $c$ on $M.$ We get the conditions under which $T^{\ast}M $ endowed with the horizontal lift $^{H}J$ of an almost complex structure $J$ and with the metric $\widetilde{g}_{\nabla,c}$ is a K\"{a}hler-Norden manifold. Also curvature properties of the Levi-Civita connection and another metric connection of the metric $\widetilde{g}_{\nabla,c}$ are presented.

Properties of Modified Riemannian Extensions

Properties of modified Riemannian extensions

Let M be an n-dimensional differentiable manifold equipped with a torsion-free linear connection $$\nabla $$
 and $$T^{*}M$$
 its cotangent bundle. The present paper aims to study a metric connection $$\widetilde{ \nabla }$$
 with nonvanishing torsion on $$T^{*}M$$
 with modified Riemannian extension $${}\overline{g}_{\nabla ,c}$$
 . First, we give a characterization of fibre-preserving projective vector fields on $$(T^{*}M,{}\overline{g} _{\nabla ,c})$$
 with respect to the metric connection $$\widetilde{\nabla }$$
 . Secondly, we study conditions for $$(T^{*}M,{}\overline{g}_{\nabla ,c})$$
 to be semi-symmetric, Ricci semi-symmetric, $$\widetilde{Z}$$
 semi-symmetric or locally conharmonically flat with respect to the metric connection $$ \widetilde{\nabla }$$
 . Finally, we present some results concerning the Schouten–Van Kampen connection associated to the Levi-Civita connection $$ \overline{\nabla }$$
 of the modified Riemannian extension $$\overline{g} _{\nabla ,c}$$
 .

On metric connections with torsion on the cotangent bundle with modified Riemannian extension

The present paper deals with the classication of innitesimal brepreserving conformal transformations on the tangent bundle, equipped with the Cheeger-Gromoll metric.

Lokman Bilen

Papers

On Kähler-Norden-Codazzi golden structures on pseudo-Riemannian manifolds

Properties of Modified Riemannian Extensions

Properties of modified Riemannian extensions

On metric connections with torsion on the cotangent bundle with modified Riemannian extension

On innitesimal conformal transformations with respect to the Cheeger-Gromoll metric