N ov 2 01 8 Generalized metallic pseudo-Riemannian structures (original) (raw)
Generalized metallic pseudo-Riemannian structures
arXiv: General Mathematics, 2018
We generalize the notion of metallic structure in the pseudo-Riemannian setting, define the metallic Norden structure and study its integrability. We construct a metallic natural connection recovering as particular case the Ganchev and Mihova connection, which we extend to a metallic natural connection on the generalized tangent bundle. Moreover, we construct metallic pseudo-Riemannian structures on the tangent and cotangent bundles.
On curvature tensors of Norden and metallic pseudo-Riemannian manifolds
Complex Manifolds, 2019
We study some properties of curvature tensors of Norden and, more generally, metallic pseudo-Riemannian manifolds. We introduce the notion of J-sectional and J-bisectional curvature of a metallic pseudo-Riemannian manifold (M, J, g) and study their properties.We prove that under certain assumptions, if the manifold is locally metallic, then the Riemann curvature tensor vanishes. Using a Norden structure (J, g) on M, we consider a family of metallic pseudo-Riemannian structures {Ja,b}a,b∈ℝ and show that for a ≠ 0, the J-sectional and J-bisectional curvatures of M coincide with the Ja,b-sectional and Ja,b-bisectional curvatures, respectively. We also give examples of Norden and metallic structures on ℝ2n.
Conjugate Connections and their Applications on Pure Metallic Metric Geometries
arXiv (Cornell University), 2022
Let (M, J, g) be a metallic pseudo-Riemannian manifold equipped with a metallic structure J and a pseudo-Riemannian metric g. The paper deals with interactions of Codazzi couplings formed by conjugate connections and tensor structures. The presence of Tachibana operator and Codazzi couplings presented a new characterization for locally metallic pseudo-Riemannian manifold. Also, a necessary and sufficient condition a non-integrable metallic pseudo-Riemannian manifold is a quasi metallic pseudo Riemannian manifold is derived. Finally, it is introduced metallic-like pseudo-Riemannian manifolds and presented some results concerning them.
General Natural Metallic Structure on Tangent Bundle
Iranian Journal of Science and Technology Transaction A-science, 2018
In this paper, we study the natural lift types as complete and horizontal of a metallic structures on the tangent bundle of a Riemannian manifold and find some interesting results about structures, metric, distribution and the relationship between them. In the end, we introduce general natural lifts of a metallic structure and a semi-Riemannian manifold on the tangent bundle and get the conditions under which the tangent bundle endowed with such a metallic structure and a general natural lifted metric is a Riemannian manifold.
Generalized metallic structures
Revista de la Unión Matemática Argentina, 2020
We study the properties of a generalized metallic, a generalized product and a generalized complex structure induced on the generalized tangent bundle of a smooth manifold M by a metallic Riemannian structure (J, g) on M , providing conditions for their integrability with respect to a suitable connection. Moreover, by using methods of generalized geometry, we lift (J, g) to metallic Riemannian structures on the tangent and cotangent bundles of M , underlying the relations between them.
A pseudo-Riemannian metric on the tangent bundle of a Riemannian manifold
Balkan Journal of Geometry and Its Applications
On the tangent bundle of a Riemannian manifold (M;g) we consider a pseudo-Riemannian metric deflned by a symmetric tensor fleld c on M and four real valued smooth functions deflned on (0;1). We study the conditions under which the above pseudo-Riemannian manifold has constant sectional curvature.
Α-Connections in Generalized Geometry
Journal of Geometry and Physics, 2021
We consider a family of α-connections defined by a pair of generalized dual quasistatistical connections (∇,∇ *) on the generalized tangent bundle (T M ⊕ T * M,ȟ) and determine their curvature, Ricci curvature and scalar curvature. Moreover, we provide the necessary and sufficient condition for∇ * to be an equiaffine connection and we prove that if h is symmetric and ∇h = 0, then (T M ⊕ T * M,ȟ,∇ (α) ,∇ (−α)) is a conjugate Ricci-symmetric manifold. Also, we characterize the integrability of a generalized almost product, of a generalized almost complex and of a generalized metallic structure w.r.t. the bracket defined by the α-connection. Finally we study α-connections defined by the twin metric of a pseudo-Riemannian manifold, (M, g), with a non-degenerate g-symmetric (1, 1)-tensor field J such that d ∇ J = 0, where ∇ is the Levi-Civita connection of g.
Almost Bronze Structures on Differentiable Manifolds
Journal of Mathematics
This study introduces a novel structure that is not included in the metallic structure family. This new structure, which is called an almost bronze structure, has been defined using a 1,1 type tensor field φ which fulfills the requirement φ 2 = m φ − I d on a differentiable manifold. We investigated the parallelism and integrability conditions of these almost bronze structures by use of an almost product structure corresponding to them. Also, we have defined an almost bronze Riemannian manifold.
A class of metrics on tangent bundles of pseudo-Riemannian manifolds
2011
We provide the tangent bundle TM of pseudo-Riemannian manifold (M, g) with the Sasaki metric gs and the neutral metric gn. First we show that the holonomy group Hs of (TM, gs) contains the one of (M, g). What allows us to show that if (TM, gs) is indecomposable reducible, then the basis manifold (M, g) is also indecomposable-reducible. We determine completely the holonomy group of (TM, gn) according to the one of (M, g). Secondly we found conditions on the base manifold under which (TM, gs) ( respectively (TM, gn) ) is Kählerian, locally symmetric or Einstein manifolds. (TM, gn) is always reducible. We show that it is indecomposable if (M, g) is irreducible.
Journal of Mathematics
We explore “the horizontal lift” of the structure J satisfying J 2 − α J − β I = 0 and establish that it as a kind of metallic structure. An analysis of Nijenhuis tensor of metallic structure J H is presented, and a new tensor field J ˜ of 1,1 -type is introduced and demonstrated to be metallic structure. Some results on the Nijenhuis tensor and the Lie derivative of J ˜ in TM are proved and explicit examples are given. Moreover, the metallic structure J ˜ endowed with projection operators l ˜ and m ˜ in TM is studied.