Almost complex structures on the tensor bundles (original) (raw)
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On almost complex structures in the cotangent bundle
Turkish Journal of Mathematics
E. M. Patterson and K. Yano studied vertical and complete lifts of tensor fields and connections from a manifold Mn to its cotangent bundle T * (Mn) . Afterwards, K. Yano studied the behavior on the crosssection of the lifts of tensor fields and connections on a manifold Mn to T * (Mn) and proved that when ϕ defines an integrable almost complex structure on Mn , its complete lift C ϕ is a complex structure. The main result of the present paper is the following theorem: Let ϕ be an almost complex structure on a Riemannian manifold Mn . Then the complete lift C ϕ of ϕ , when restricted to the cross-section determined by an almost analytic 1 -form ω on Mn , is an almost complex structure.
Proposed theorems for lifts of the extended almost complex structures on the complex manifold
arXiv (Cornell University), 2020
It is well known that the tensor field J of type (1,1) on the manifold M is an almost complex structure if J 2 = −I, I is an identity tensor field and the manifold M is called the complex manifold. Let k M be the k order extended complex manifold of the manifold M. A tensor field J k on k M is called extended almost complex structure if (J k) 2 = −I. The present paper aims to study the higher order complete and vertical lifts of the extended almost complex structures on an extended complex manifold k M. The proposed theorems on the Nijenhuis tensor of an extended almost complex structure J k on the extended complex manifold k M are proved. Also, a tensor fieldJ k of type (1,1) is introduced and shows that it is an extended almost complex structure. Finally, the Lie derivative concerning higher-order lifts is studied and basic results on the almost analytic complex vector concerning an extended almost complex structure on k M are investigated.
The Geometry of Tangent Bundles and Almost Complex Structures
2018
In this paper, we study the geometry of a tangent bundle of a Riemannian manifold endowed with a Sasaki metric. Using O’Neill tensors given in [7], we prove some characteristic theorems comparing the geometries of a smooth manifold and its tangent bundle. We also show that there exists an almost complex structure on a Riemannian manifold which is not holomorphic to the canonical almost complex structure of its tangent bundle.
Almost complex structures on cotangent bundles and generalized geometry
Journal of Geometry and Physics, 2010
We study a class of complex structures on the generalized tangent bundle of a smooth manifold M endowed with a torsion free linear connection, ∇. We introduce the concept of ∇-integrability and we study integrability conditions. In the case of the generalized complex structures introduced by Hitchin (2003) in [2], we compare the two concepts of integrability. Moreover, as an application, we describe almost complex structures on the cotangent bundle of M induced by complex structures on the generalized tangent bundle of M.
arXiv: Differential Geometry, 2020
In this article, we study higher-order complete lifts of the tensor field of type (1,1) and find some theorems on Nijenhuis tensor. We also consider an almost complex structure on complex manifold M and defined extended almost complex structures on ^kM. Moreover, we study Lie derivative concerning higher-order lifts and find some basic results on the almost analytic complex vector concerning an extended almost complex structure on ^kM.
On Certain Structures Defined on the Tangent Bundle
Rocky Mountain Journal of Mathematics, 2006
The differential geometry of tangent bundles was studied by several authors, for example: Davies [4], Yano and Davies [5], Dombrowski [6], Ledger and Yano [9] and Blair [1], among others. It is well known that an almost complex structure defined on a differentiable manifold M of class C ∞ can be lifted to the same type of structure on its tangent bundle T (M). However, when we consider an almost contact structure, we do not get the same type of structure on T (M). In this case we consider an odd dimensional base manifold while our tangent bundle remains to be even dimensional. The purpose of this paper is to examine certain structures on the base manifold M in relation to that of the tangent bundle T (M).