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

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).

Calibrated complex structures on the generalized tangent bundle of a Riemannian manifold

Journal of Geometry and Physics, 2006

We study calibrated complex structures on the generalized tangent bundle of a Riemannian manifold M and their relationship to the Riemannian geometry of M. In particular we introduce a concept of integrability of such structures and we prove that integrability conditions are strictly related to the existence of certain Codazzi tensors on M.

Pseudo-Hermitian immersions, pseudo-Einstein structures, and the Lee class of a CR manifold

Kodai Mathematical Journal, 1996

Any nondegenerate CR manifold carrying a fixed contact 1-form is known to possess (cf. N. Tanaka [T], S. Webster [Wl]) a canonical linear connection (the Tanaka-Webster connection) parallelizing the Levi form and the maximal complex structure. This leads to an (already widely exploited, cf. D. Jerison & J.M. Lee [JL1], [JL2], J.M. Lee [LI], [L2], H. Urakawa [Ul], [U2], etc.) analogy between CR geometry on one hand, and both Hermitian and conformal geometry on the other.

CR-submanifolds of a Kaehler manifold. I

Journal of Differential Geometry, 1981

BANG-YEN CHEN 1. Introduction Let M be a Kaehler manifold with complex structure /, N a Riemannian manifold isometrically immersed in M, and 6 ΰ x the maximal holomorphic subspace of the tangent space T X N of N. If the dimension of β ύ x is the same for all x in N, 6 ί) x gives a holomorphic distribution ^D on N. Recently, A. Bejancu [1] introduced the notion of a CR-submanifold of M as follows. A submanifold N in a Kaehler manifold M is called a CΛ-submanifold if there exists oniVa differentiable holomorphic distribution ty such that its orthogonal complement 6 ί} 1 is a totally real distribution, i.e., In this series of papers, we shall obtain some fundamental properties of CΛ-submanifolds in Kaehler manifolds. 2. Preliminaries Let M be a complex w-dimensional Kaehler manifold with complex structure /, and N a real n-dimensional Riemannian manifold isometrically immersed in M. We denote by < ,) the metric tensor of M as well as that induced on N. Let V and V be the covariant differentiations on N and M, respectively. Then the Gauss and Weingarten formulas for N are given respectively by (2.1) v^y=v^y+σ(^,y), (2.2) V x ξ =-A ζ X+D x t for any vector fields X y Y tangent to N and any vector field ξ normal to N 9 where σ denotes the second fundamental form, and D the linear connection, called the normal connection, induced in the normal bundle T ± N. The second fundamental tensor A± is related to σ by (2.3) (AtX, 7> = <σ(*. Y), «>.

CR submanifolds of maximal CR dimension in complex manifolds

PDEs, Submanifolds and Affine Differential Geometry, 2002

Let M be a CR submanifold of maximal CR dimension in a complex projective space such that the distinguished vector field ξ is parallel with respect to the normal connection. In this article we treat the special case when the shape operator with respect to this vector field has exactly two distinct eigenvalues and we give another sufficient condition for M to be an open subset of a geodesic sphere by discussing its holomorphic sectional curvature.