CR− structures on the Unit Cotangent Bundle and Bochner Type Tensor (original) (raw)

Codimension reduction and second fundamental form of CR submanifolds in complex space forms

Journal of Mathematical Analysis and Applications, 2009

Considering n-dimensional real submanifolds M of a complex space form (M n+p ,ḡ, J) which are CR submanifolds of CR dimension n−m 2 , we study the condition h(F X, Y) + h(X, F Y) = 0 on the structure tensor F naturally induced from the almost complex structure J of the ambient manifold and on the second fundamental form h of submanifolds M.

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.

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.

A remark on almost complex manifold with linear connections

INTERNATIONAL SCIENTIFIC AND PRACTICAL CONFERENCE “TECHNOLOGY IN AGRICULTURE, ENERGY AND ECOLOGY” (TAEE2022)

The complex and almost complex manifolds are enormous and very fruitful fields for differential geometry. J.A. Schouten and D. Van Dantzig were the first to try to apply the finding in differential geometry of spaces with Riemannian metric and affine connection to the situation of complex structure spaces. C. Ehresmann defined an almost complex space as an even-dimensional differentiable manifold containing a tensor field with a square root of minus unity. The present paper intended to study, some fundamental properties with linear connections of an almost complex manifold. If almost complex structure be converted from a complex structure, then the various integrable and completely integrable condition has been investigated. Furthermore, the symmetric affine connections of almost complex manifold have also been 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.

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).