Curvature Identities for Generalized Kenmotsu Manifolds

In the present paper we obtained 2 identities, which are satisfied by Riemann curvature tensor of generalized Kenmotsu manifolds. There was obtained an analytic expression for third structure tensor or tensor of f-holomorphic sectional curvature of GK-manifold. We separated 2 classes of generalized Kenmotsu manifolds and collected their local characterization.


Introduction
Let M be a connected smooth manifold of (2n+1) dimension, ∞ ( ) is the algebra of smooth functions on M, ( ) -∞ -module of smooth vector fields on M, d is the operator of exterior differentiation. If M preserves Riemannian metric 〈•,•〉, then corresponding Riemannian connection is expressed as . In the future, all manifolds, tensor fields (tensors, and similarly objects are assumed to be smooth of class ∞ . Differential 1-form of maximal rank on an odd-dimensional Riemannian manifold produce a special differential-geometric structure called contact metric structure that naturally generalizes to the so-called almost contact metric structure. We recall [1] that a contact form or contact structure on an odd-dimensional manifold M, dim = 2 + 1, is called 1-form  on M, which in each point of the manifold is ∧ ( ) ∧ … ∧ ( ) ⏟ ≠ 0, i.e. = dim is in each point of M. Manifold with a fixed contact form on it is called a contact manifold [2]. Distributions ℒ = ker , dim ℒ = 2 , and ℳ = ker( ) , dim ℳ = 1 are determined on such manifold internally. It is easy to derive from Darboux theorem [1] that ℒ ∩ ℳ = {0}, consequently ( ) = ℒ⨁ℳ. Let us take ∈ ℳ such that ( ) = 1. Then, it is possible to determine mutually complementing projections = ⨂ and ℓ = − ⨂ on distributions ℳ and ℒ respectively. Let Riemannian metric h be fixed on M. Based on the metric ℎ|ℒ, it is not difficult to build metric 〈•,•〉 on ℒ such that the operator : ℒ → ℒ , which is determined with the identity 〈 , 〉 = ( , ); , ∈ ℒ, will be involutory, i.e. 2  (1.1) A contact manifold M that is provided with Riemannian metric 〈•,•〉, for which relations (1) are true, is called a contact metric manifold.
The given construction makes the following definition natural. Definition 1.1 [3]. Almost a contact metric (or almost Gray's) structure on a manifold M is a set of { , , , } tensor fields on M, where = 〈•,•〉 is a (pseudo-) Riemannian metric, f is a tensor of (1.1) kind called structure endomorphism, ξ is a vector field, which is called characteristic,  is a differential 1-form, which is called a contact form of structure. Here:   [3] as such. Its assignment matches the assignment of Gstructure on M with the structural group * × ( , ), where * is a multiplicative group of positive real numbers. In fact, this is how J. Gray introduced it in 1959 [2]. Assignment of the almost contact structure {Φ, , } on manifold M induces a canonical hyper-distribution ℒ = ker on this manifold, called contact, which is invariant in relation to f. Due to (1.2:4), operator f induces an almost complex structure on ℒ. This gave grounds for J. Bouzon to assume an almost contact structure as an almost complex [5]. If { , , , } is an almost contact metric structure, then by the virtue of (1.2:5), the pair { |ℒ, |ℒ} sets an almost Hermitian structure on this hyper-distribution, due to which an almost contact metric structure can be naturally called a metric almost cocomplex structure. Obviously, its assignment equally matches the assignment of G-structure on M with a structural group { } × ( ). We note that the concept of an almost contact structure has historically evolved in the following sequence: S.S. Chern [6] found that contact manifold admits G-structure with structural group { } × ( ); J. Gray called manifolds that admit such structure as almost contact manifolds [2]. S. Sasaki noticed that such a G-structure generates a triple { , , } that has the above mentioned conditions (1.2:3) and (1.2:4), from which it is to derive (1.2:1) and (1.2:2). Moreover, based on an arbitrary Riemannian metric h on such manifold, he constructed Riemannian metric 〈 , 〉 = ℎ( , ) + ℎ( 2 , 2 ) + ( ) ( ) , complementing { , , } to almost contact metric structure [3].
The most important example of almost contact metric structures, which largely determines their role in differential geometry, is the structure induced on the hypersurface N of the manifold M equipped with an almost Hermitian structure { , 〈•,•〉}. Let us recall this construction. Let 0 be a unit normal field to N. Then vector is = ( 0 ) ∈ ( ), where its orthogonal complement on N is invariant to J. We define in ( ) a linear operator Φ = | ⨁0| , where is the linear hull of the vector ξ, and 1-form is ( ) = 〈 , 〉. Then {〈•,•〉, , , } is the almost contact metric structure on N. In particular, such a structure is induced on the odd-dimensional sphere 2 −1 considered as a hypersurface in the realification of the space . This is the most important and, apparently, historically the first concrete example of such a structure.
Then { = 〈•,•〉, , , } is the almost contact metric structure on manifold M. It is well-known [1] that in this case almost Hermitian structure { , ℎ} induces on the manifold × , where = | ⨁ 1 , ℎ = | ⨁ 1 , 1 is a canonical almost complex structure on the 2-dimensional distribution × , 1 is a metric on this distribution being a direct sum of metric | and the canonical metric on R. An almost contact structure { , , } is called normal if the structure { , ℎ} is integrable [4]; a necessary and sufficient condition for the structure to be normal has the form + 1 2 ⨂ = 0, where N is the Nijenhuis tensor of the operator f [4].
Today, there active studies of the geometry of almost contact metric structures on manifolds. One of the most pressing issues in this section of geometry is the study of individual classes of almost contact metric manifolds. In 1972, Kenmotsu [7] introduced a class of almost contact metric structures characterized by an identity ∇ ( ) = 〈 , 〉 − ( ) ; , ∈ ( ) . Kenmotsu structures, for example, naturally arise in the Tanno classification of connected almost contact metric manifolds such that automorphism group has maximum dimension [8]. (see [7]). Polarizing the identity characterizing the Kenmotsu manifolds, S.V. Umnova [10] identified in his thesis paper a class of almost contact metric manifolds; this class was a generalization of Kenmotsu manifolds and was called the class of generalized (in short, GK-) Kenmotsu manifolds. In [10], it was proved that generalized Kenmotsu manifolds of constant curvature are Kenmotsu manifolds of constant curvature −1.
In [11], this class of manifolds is called as a class of nearly Kenmotsu manifolds. The authors prove that a second-order symmetric closed recurrent tensor, recurrence covector of which annihilates the characteristic vector ξ, is a multiple of the metric tensor g. In addition, the authors consider the f-recurrent nearly Kenmotsu manifolds. It is proved that frecurrent nearly Kenmotsu manifolds are Einstein manifolds, and locally f-recurrent nearly Kenmotsu manifolds are manifolds of constant curvature −1.
In [12], M.B. Banaru studied hypersurfaces of almost Hermitian manifolds of class 3 with the Kenmotsu structure and obtained interesting properties of Kenmotsu manifolds. In our papers [13][14], Einstein's generalized Kenmotsu manifolds were studied; contact analogs of Gray identities were obtained, three classes of this type of manifolds were distinguished; a local characterization of the distinguished classes of manifolds was obtained. In [15], the curvature identities for the Riemannian curvature tensor were considered for the particular case of generalized Kenmotsu manifolds, called special generalized Kenmotsu manifolds of second kind [10].
The paper [16] examines the integrability properties of generalized Kenmotsu manifolds. In this paper, we investigate GK-manifolds, the first fundamental distribution of which is completely integrable. It is shown that an almost Hermitian structure induced on integral manifolds of maximum dimension of the first distribution of a GK-manifold is nearly Kahlerian. Local structure of a GK-manifold with a closed contact form is obtained, expressions for the first and second structure tensors are given. The components of the Nijenhuis tensor of a GK-manifold are also calculated. Since defining the Nijenhuis tensor is equivalent to defining four tensors (1) , (2) , (3) , (4) , the geometric meaning of vanishing of these tensors is studied. Local structure of an integrable and normal GK-structure is obtained. It is proved that characteristic vector of GK-structure is not a Killing vector.
It is clear from the given reviews of works on generalized Kenmotsu manifolds that the interest in studying of this class of manifolds does not fade, but rather grows.
In this paper, we continue the study of generalized Kenmotsu manifolds and investigate the geometry of the Riemannian curvature tensor for this class of manifolds.
This paper is organized as follows. In paragraph 2, we give preliminary information needed in the further presentation and construct the space of the adjoint G-structure. In paragraph 3, we give a definition of generalized Kenmotsu manifolds, provide a complete group of structure equations, and give the components of the Riemann-Christoffel tensor on space of the adjoint G-structure. In paragraph 4 we, using the procedure of restoring the identity of [17][18], obtain some identities, which are satisfied by the Riemannian curvature tensor of generalized Kenmotsu manifolds and, on their basis, we separate two classes of generalized Kenmotsu manifolds. In addition, we get a local characterization of these classes.

Preliminaries
Let M be a smooth manifold of dimension 2 + 1, ( ), C ∞ be a module of smooth vector fields on manifold M. Further, all manifolds, tensor fields and similarly objects are assumed to be smooth of class C ∞ .
Definition 2.1 ( [17][18]). Almost contact structure on a manifold M is a triple ( , , ) tensor fields on this manifold, where η is a differential 1-form called the contact form of structure, is a vector field called characteristic, f is endomorphism of module ( ) called the structure endomorphism. Here The mappings : ℒ ⟶ √−1 and ̅ : ℒ ⟶ −√−1 are, respectively, isomorphisms and anti-isomorphisms of Hermitian spaces. Therefore, to each point ∈ 2 +1 one can attach a family of frames of the space ( ) of the form ( , 0 , 1 , … , , 1 , … ,̂), where = √2 ( ),̂= √2 ̅ ( ), 0 = ; where { } is an orthonormal basis of Hermitian space ℒ . This frame is called an A-frame [18]. It is easy to see that matrices of the tensor components and in the A-frame have the form: where is the identity matrix of size n. It is well-known [17,18]  On top of that, note that, since the corresponding forms and tensors are real, ̅̅̅ =, ̅ =̂, ∇ , ̅̅̅̅̅̅ = ∇̂,̂, where → ̅ is the complex conjugation operator.
The first group of structure equations for Riemannian connection = − ∧ on the space of the adjoint G-structure of an almost contact metric manifold, can be written in the following form, called the first group of structure equations for an almost contact metric manifold [17][18]
The following theorem takes place.
Theorem 3.1 [13]. The complete group of structure equations for GK-manifolds on the space of the adjoint G-structure has the form: Definition 3.2 [10]. GK-structure is called: a special generalized Kenmotsu structure of the first kinds (in short SGK-structure of the first kind) if C dbc = C dbc = 0 ; a special generalized Kenmotsu structure of the second kind (in short SGK-structure of the second kind) if = = 0. Identity is called the first fundamental identity. Identity is called the second fundamental identity. Identity is called the third fundamental identity. Let M be a GK-manifold. Let us recall the following theorems from [13][14]. Theorem 3.2 [13]. The nonzero essential components of the Riemann-Christoffel tensor on the space of the adjoint G-structure have the form:

Curvature identities for GK-manifolds
In [19], we obtained several identities for the Riemannian curvature tensor of generalized Kenmotsu manifolds and separated two subclasses of GK-manifolds. In addition, there were obtained the local structure of separated classes for GK-manifolds. In [20], two classes of generalized Kenmotsu manifolds were separated, called the class of f-holomorphic and the class of f-paracontact manifolds; a complete classification of the separated classes was obtained. In this paragraph we will also, as in [19] and [20], consider two classes of generalized Kenmotsu manifolds.