Tensor invariants of generalized Kenmotsu manifolds

In this paper, we study the properties of generalized Kenmotsu manifolds, consider the second-order differential geometric invariants of the Riemannian curvature tensor of generalized Kenmotsu manifolds (by the symmetry properties of the Riemannian geometry tensor). The concept of a tensor spectrum is introduced. Nine invariants are singled out and the geometric meaning of these invariants turning to zero are investigated. The identities characterizing the selected classes are singled out. Also, 9 classes of generalized Kenmotsu manifolds are distinguished, the local structure of 8 classes from the selected ones is obtained.


Introduction
Let M be a connected smooth manifold of dimensions (2n+1), ∞ ( ) the algebra of smooth functions on M, ( ) -∞ -module of smooth vector fields on the manifold M, d an outer derivation operator. If a Riemannian metric 〈•,•〉 is given on M, then the corresponding Riemannian connection will be marked by the symbol . All manifolds, tensor fields (tensors), etc. objects are assumed to be smooth of class ∞ . Differential 1-forms of maximum rank on an odd-dimensional Riemannian manifold generate a special differential-geometric structure, called a contact metric structure, which naturally generalizes to a so-called almost contact metric structure.
A lot of works are devoted to the study of almost contact metric structures. We have chosen for a study a rather interesting class of almost contact metric structures, called the class of generalized Kenmotsu manifolds.
Generalized Kenmotsu manifolds were first introduced as approximately trans-Sasakian manifolds in the study [1]. In this paper discloses relations in this class of manifolds. In addition, some new classes of almost contact metric manifolds are introduced and the inclusion relations of these classes with generalized Kenmotsu manifolds are presented. It is shown that normal generalized Kenmotsu varieties are Kenmotsu varieties.
Further insight into the geometry of generalized Kenmotsu manifolds is carried out in the research paper by S.V. Umnova [2], where the definition of generalized (in short, GK-) Kenmotsu manifolds is given. In work [2] Umnova S.V. distinguishes two subclasses of generalized Kenmotsu varieties, called special generalized Kenmotsu varieties (for short, In the paper [19], it is proved that a GK-manifold of dimension other than 5 is an SGKmanifold of the second type. Consequently, we get that a GK-manifold, which is an SGKmanifold of the first type, has dimension 5. Paper [20] investigates the integrability properties of generalized Kenmotsu manifolds. The first, second and third fundamental identities of GK-structures are defined. 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 Kähler. The local structure of a GK-manifold with a closed contact form is obtained, expressions for the first and second structure tensors are given. The geometric meaning of the vanishing of the tensors (1) , (2) , (3) , (4) is investigated. The local structure of an integrable and normal GKstructure is obtained. It is proved that the characteristic vector of the GK-structure is not a Killing vector.
From the above historical survey of works devoted to generalized Kenmotsu varieties, it is clear, that the interest to such studies does not fade away, though grows.
In this paper, we continue the study of generalized Kenmotsu manifolds and investigate second-order differential geometric invariants of the Riemannian curvature tensor of generalized Kenmotsu manifolds.
This paper is structured as follows. In Section 1, we give preliminary information needed in the further presentation: we construct the space of the associated G-structure of an almost contact metric manifold; we write down the first group of structural equations of an almost contact metric manifold on the space of the associated G-structure; we introduce structural tensors of an almost contact metric structure and present their properties.
In Section 2, we give a definition of generalized Kenmotsu manifolds, give a complete group of structure equations, and give five fundamental identities of generalized Kenmotsu manifolds. In Section 3, we present the components of the Riemann-Christoffel tensor on the space of the associated G-structure, and introduce the concept of the tensor spectrum. Nine tensor invariants of generalized Kenmotsu manifolds are distinguished, identities characterizing these invariants are obtained, and 9 classes of generalized Kenmotsu manifolds are distinguished. In addition, we obtain a local characterization of 8 of the selected classes.

Preliminary information
Let M be a smooth manifold of dimension 2 + 1.
Definition 1.1 [21]. An almost contact structure on a manifold M is a triple ( , , Φ) of tensor fields on this manifold, where η is a differential 1-form called the contact form of the structure, ξ is a vector field called characteristic, Φ is an endomorphism of the module ( ) called structural endomorphism. Herewith A manifold on which an almost contact (metric) structure is fixed is called an almost contact (metric (in short, AC-)) manifold.
The complexification ( ) of the module ( ) decomposes into a direct sum Displayed values : ℒ ⟶ ( Φ √−1 ) and σ ̅ : ℒ ⟶ ( Φ −√−1 ) are isomorphisms and antiisomorphisms, respectively, 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 the Hermitian space ℒ . Such a frame is called an A-frame [21]. It is easy to see that the matrices of the components of the tensors Φ and in the A-frame have the form, respectively: where is the identity matrix of order n. It is well known [21] that the set of such frames defines a G-structure on M with the structure group {1} × ( ) represented by matrices of the form ( 1 0 0 0 0 0 0 ), where ∈ ( ). This G-structure is called adjoint [21].
Let ( 2 +1 , Φ, , , = 〈•,•〉) be an almost contact metric manifold. Let us agree that throughout the entire work, unless otherwise stated, the indices , , , , … run through the values from 1 to 2 , the indices , , , , … are values from 1 to n, and put ̂= + ,̂= , 0 = 0. Let ( , ) be a local map on a manifold M. According to the Basic Theorem of tensor analysis [22, p. 243], the assignment of a structural endomorphism Ф and a Riemannian structure = 〈⋅,⋅〉 on a manifold M induces the assignment of a bundle on the total space BM frames over M of a system of functions {Φ }, { } satisfying in the coordinate neighborhood = −1 ( ) ⊂ a system of differential equations of the form where { }, { } are the components of the displacement forms and the Riemannian connection , respectively, Φ , , , are the components of the covariant differential of the tensors Φ and g in this connection, respectively. Moreover, by the definition of the Riemannian connection, ∇ = 0 and, therefore, Relations (1.4) on the space of the associated G-structure can be written in the form [21,22] Φ , = 0, Φ̂, = 0, Φ 0, In addition, note that, since the corresponding forms and tensors are real, ̅̅̅ =, ̅ =̂, ∇Φ , ̅̅̅̅̅̅̅ = ∇Φ̂,̂, where → ̅ is the operator of complex conjugation.
The first group of structural equations of Riemannian connection = − ∧ , on the space of the associated 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 [21,22]: where = 0 = * ( ); π is the natural projection of the space of the adjoint G-structure onto the manifold M, = , We introduce the notation [20] Consider the following families of functions on the space of the associated G-structure These systems of functions define tensors of the corresponding types on the manifold M, which are called the first, second, ..., sixth structural tensors of the AC-structure, respectively. The following takes place Proposition 1.1 [22]. Structural tensors of the AC-structure have the following properties:
, [9], [12]). The class of almost contact metric manifolds characterized by the identity are called generalized Kenmotsu manifolds (in short, GK-manifolds). Note that in print this class of manifolds is called the nearly Kenmotsu class of manifolds ( [3 -9] and others). We will call these manifolds, as in [2], generalized Kenmotsu manifolds, and briefly GK-manifolds.
The following theorem holds. Theorem 2.1 [19]. The complete group of structure equations for GK-manifolds on the space of the associated G-structure has the form: For generalized Kenmotsu varieties, the following fundamental identities hold [23]. Identity will be called the first fundamental identity. Identity will be called the second fundamental identity. Identity will be called the third fundamental identity. Identity 2F ab F cd = F ac F db + F ad F bc ; (2.7) will be called the fourth fundamental identity. Identity C abcg C gdh = 0 (2.8) will be called the fifth fundamental identity of GK-manifolds. Relations (1.6) for GK-manifolds take the form: ,̂= , defined on the space of the associated G-structure, defines a globally defined tensor of type (1,1) on the manifold M, which is called the second structure tensor of the GK-structure.
The first structure tensor of GK-manifolds has the following form: The second structure tensor of GK-manifolds has the form: ( ) = −∇ ; ∈ ( ). (2.12) Definition 2.2 [2,12]. A GK-structure is called a special generalized Kenmotsu structure of the first kind (in short, an SGK-structure of the first kind) if C abc = C abc = 0; a special generalized Kenmotsu structure of the second kind (in short, the SGK-structure of the second kind), if = = 0. If = = 0 and = = 0, then the GK-structure is a Kenmotsu structure.

Identities of curvature of GK-manifolds
Let M be a GK-variety. We recall the following theorem from ( [13], [19]).  The set of 3 +1 tensors of type ( , 1) in the expansion (3.2) is called the spectrum of the tensor T, and these tensors themselves are called the elements of the spectrum. The element of the tensor spectrum is characterized by a decimal number, in which ternary units correspond to the numbers of linear arguments, two correspond to the numbers of antilinear arguments, and zeros correspond to the numbers of self-adjoint arguments. For example, ⏟ (16) characterizes the element of the spectrum of the tensor T, linear in the first and third arguments and antilinear in the second argument, since 16 = 1 • 3 2 + 2 • 3 1 + 1 • 3 0 .
The proof follows directly from the symmetry properties of the Riemannian curvature tensor, Definition 3.1 and Theorem 3.4.
Remark. Since, by virtue of the symmetry properties of the Riemannian curvature tensor, from 00 = 0 it follows that 00̂= 0, that is, a GK-variety of class ℛ 1 is a variety of class ℛ 3 . The converse is also true.
According to Theorem 3.4, we call the invariants ℛ 1 , ℛ 3 , ℛ 6 , ℛ 8 , ℛ 9 the basic invariants of the GK-manifold. The following theorem gives explicit expressions for the main invariants of a GK-manifold. Theorem 3.5. 1) The invariant 5 of the GK-manifold is calculated by the formula 2) The invariant ℛ 6 of the GK-manifold is calculated by the formula 3) The invariant ℛ 8 of the GK-manifold is calculated by the formula 4) The invariant ℛ 9 of the GK-manifold is calculated by the formula Proof. We carry out the proof for case 1). The rest of the cases are proved similarly. Taking into account (1.3) and (3.1), we have: Thus, So, Similarly, we obtain Thus, the functions { } and {̂̂̂} are components of the tensor and therefore this tensor coincides with the tensor 5 ( , ) , as required Theorem 3.6. A GK-manifold of class 5 is a Kenmotsu variety, that is, a manifold obtained from a cosymplectic manifold by a canonical concircular transformation of a cosymplectic structure.
Proof. Let M be a GK-manifold of class 5 , that is, 5  = 0 ⟺ = 0. Thus, a GK-manifold of class 5 is a Kenmotsu manifold. Therefore, according to [24], it is a manifold obtained from a cosymplectic manifold by a canonical concircular transformation of a cosymplectic structure. Theorem 3.7. A GK-manifold of class 6 is a special generalized Kenmotsu manifold of the second type.
Proof. Let M be a GK-manifold of class 6 , that is, 6  Using the results on SGK-manifolds of the second type obtained in [2], [19], Theorem 3.7 can be formulated as: Theorem 3.8. A GK-manifold of class 6 coincides with an almost contact metric manifold obtained from the most exact cosymplectic manifold by the canonical concircular transformation of the most exact cosymplectic structure. . We contract the resulting identity with the object , then, taking into account the first fundamental identity, we obtain = − . We contract the resulting equality by indices c and f, then we get ∑ | | 2 = 2 . (3.8) The last equality implies: 1) = 0, i.e. the manifold M is an SGK-manifold of the second type; 2) ∑ | | 2 = 2.
In the second case, consider the third fundamental identity, i.e. It follows from (3.9) and (3.10) that [ | | ] = 0. The resulting equality, due to the fourth fundamental identity, can be written in the form = 0, which, due to the condition ∑ | | 2 = 2, will be written in the form 2 = 0. Thus, = 0, i.e. the manifold is a special generalized Kenmotsu manifold of the second type.
Taking into account that an SGK-structure of the second type is obtained by the canonical concircular transformation of the most exact cosymplectic structure, and using the local structure of the most exact cosymplectic manifold [25], Theorem 3.9 can be formulated as follows: Theorem 3.10. A simply connected GK-manifold of class 8 is canonically concircular to the product of an approximately Kähler manifold and the real line.