Geometry of Harmonic Nearly Trans-Sasakian Manifolds

Abstract

This paper considers a class of nearly trans-Sasakian manifolds. The local structure of nearly trans-Sasakian structures with a closed contact form and a closed Lee form is obtained. It is proved that the class of nearly trans-Sasakian manifolds with a closed contact form and a closed Lee form coincides with the class of almost contact metric manifolds with a closed contact form locally conformal to the closely cosymplectic manifolds. A wide class of harmonic nearly trans-Sasakian manifolds has been identified (i.e., nearly trans-Sasakian manifolds with a harmonic contact form) and an exhaustive description of the manifolds of this class is obtained. Also, examples of harmonic nearly trans-Sasakian manifolds are given.

1. Introduction

The geometrical properties of almost Hermitian and almost contact metric structures have a number of interesting relationships. Thus, the most important example of almost contact metric structures, which largely determines their role in differential geometry, is the structure induced on the hypersurface of an almost Hermitian manifold. On the other hand, if $\left\{\mathrm{\Phi },\xi ,\eta ,g=⟨·,·⟩\right\}$ is an almost contact metric structure on the manifold M, it is well-known [1] that an almost Hermitian structure $\{J,h\}$ can be canonically induced in the manifold $M\times \mathbf{R}$, where $J=\mathrm{\Phi }|\mathcal{L} \oplus J_1$, $h=g|\mathcal{L} \oplus g_1$, and $J_1$ is a canonical almost complex structure on a two-dimensional distribution $ker\left(d\eta \right)\times \mathbf{R}$, $g_1$ is a metric on this distribution, which is the direct sum of the metric $g\left|ker\right(d\eta )$ and the canonical metric $\mathbf{R}$, $\mathcal{L}=ker\eta$ (called the linear expansion of the original almost contact metric structure [2]). The problem of the relationship between these structures has been studied many times. Thus, Oubina [3] identified the classes of trans-Sasakian and almost trans-Sasakian structures, linear extensions of which belong to the classes $W_4$ and $W_2\oplus W_4$ of almost Hermitian structures in Gray–Hervella classification [4], respectively.

In [2], a class of special almost trans-Sasakian manifolds was singled out and it was proved that a connected almost contact metric manifold is the special almost trans-Sasakian if and only if it is either homothetic to an almost Sasakian manifold or locally conformal to an almost cosymplectic manifold and has a closed structural form. The most important example of special almost trans-Sasakian manifolds are trans-Sasakian manifolds, i.e., almost contact metric manifolds whose linear extension belongs to the class $W_4$ in the Gray–Hervella classification [3,4]. The following theorem can be considered as the main result of this paper: The class of trans-Sasakian manifolds with non-integrable structure coincides with the class of almost contact metric manifolds homothetic to Sasaki manifolds. The class of trans-Sasakian manifolds with an integrable structure coincides with the class of normal almost contact metric manifolds locally conformal to cosymplectic manifolds.

In [5], almost contact metric structures were considered, which are a linear extension of almost Hermitian structures of the class $W_1\oplus W_4$ in Gray–Herwell classification. Such structures are called nearly trans-Sasakian structures. In addition, in [5], the authors single out a special class of nearly trans-Sasakian manifolds. A nearly trans-Sasakian structure with a closed contact form is called the eigen nearly trans-Sasakian structure. In [5], a criterion is proved for when an almost contact metric structure is an eigen nearly trans-Sasakian structure. It is proved that the class of nearly trans-Sasakian manifolds with non-closed contact form coincides with the class of almost contact metric manifolds homothetic to Sasaki manifolds. The most important examples of such manifolds are Kenmotsu manifolds and special generalized Kenmotsu manifolds of the second kind. Theorem 3.16 from [5] is of interest: The class of nearly trans-Sasakian manifolds with a closed contact form and a closed Lee form coincides with the class of almost contact metric manifolds, with a closed contact form locally conformal to closely cosymplectic manifolds.

This result makes us investigate nearly trans-Sasakian manifolds with certain properties. For example, nearly trans-Sasakian manifolds which a structural field is a Killing vector field. Manifolds of this kind have interesting geometric properties. The geometry of these manifolds is more meaningful. In the theory of Riemannian spaces, spaces with other structures that arise in theoretical physics play an important role, especially Kähler and Sasaki spaces, and their generalizations, see [6,7,8,9,10,11,12]. The properties of the geometry of such spaces are richer in content. In this article, we are interested in the class of harmonic nearly trans-Sasakian manifolds, i.e., nearly trans-Sasakian manifolds with a harmonic contact form. The concept of harmonic nearly trans-Sasakian manifolds was introduced in [5,13]. The main purpose of this work is to study the local structure of harmonic nearly trans-Sasakian manifolds.

The structure of the work is as follows: In Section 2, we provide the preliminary information necessary for the further presentation. In this section, we define the eigen nearly trans-Sasakian structure, prove the characteristic identity of this class of structures. And we obtain the local structure of the eigen nearly trans-Sasakian structure. Here are two examples of this class of structures. In Section 3, we introduce the concepts of the harmonic nearly trans-Sasakian structure and the characteristic of the nearly trans-Sasakian structure. Examples of harmonic nearly trans-Sasakian manifolds are given. The complete classification of harmonic nearly trans-Sasakian manifolds is obtained, and a mechanism for obtaining examples of this class of manifolds is given.

2. Preliminary Information

Assume that $M^{2n+1}$ be a smooth manifold, $\mathcal{X}\left(M\right)$ is a smooth vector fields module on M, ∇ is a Riemannian connection on M, d is the de Rham differential. All manifolds, tensor fields, etc., are assumed to be smooth of class $C^{\infty }$.

Definition 1

([14]). An almost contact metric (in short AC-) structure, on the manifold M is called a set $\left(\eta ,\xi ,\mathrm{\Phi },g\right)$ tensor fields on this manifold, where η is a differential 1-form, called a contact form of the structure, ξ is a vector field, called a characteristic field, Φ is an endomorphism of the module $\mathcal{X}\left(M\right)$ called a structural endomorphism, $g=⟨·,·⟩$ is Riemannian metric. Herewith

$$\begin{array}{cc}& \left(1\right)\eta \left(\xi \right)=1;\\ & \left(2\right)\mathrm{\Phi }\left(\xi \right)=0;\\ & \left(3\right)\eta \circ \mathrm{\Phi }=0;\\ & \left(4\right){\mathrm{\Phi }}^2=-id+\xi \otimes \eta ;\\ & \left(5\right)⟨\mathrm{\Phi }X,\mathrm{\Phi }Y⟩=⟨X,Y⟩-\eta \left(X\right)\eta \left(Y\right);X,Y\in \mathcal{X}\left(M\right).\end{array}$$

(1)

A manifold on which an almost contact metric structure is fixed is called an almost contact metric (in short $AC$-) manifold.

As mentioned above, such structures naturally arise on hypersurfaces of almost Hermitian manifolds, on the main spaces $T^1$ of bundles over symplectic manifolds with an integer fundamental form (Boothby-Wang bundles) and, more generally, over almost Hermitian manifolds and are natural generalizations of the so-called contact metric manifolds arising on odd-dimensional manifolds with a fixed 1-form maximum rank (contact structure).

It is well known that a manifold that admits an $AC$-structure is odd and orientable. In a $C^{\infty }\left(M\right)$-module of smooth vector fields on such a manifold, two mutually complementary projectors

$$\begin{array}{cc}& \left(1\right)\mathbf{l}=id-\mathbf{m}=-{\mathrm{\Phi }}^2;\\ & \left(2\right)\mathbf{m}=\xi \otimes \eta \end{array}$$

(2)

are internally definite. Their images are denoted

$$\begin{array}{cc}& \left(1\right)\mathcal{L}=Im\mathrm{\Phi }=ker\eta ;\\ & \left(2\right)\mathcal{M}=ker\mathrm{\Phi },\end{array}$$

(3)

respectively. Thus, $\mathcal{X}\left(M\right)=\mathcal{L}\oplus \mathcal{M}$.

The assignment of an $AC$-structure on the manifold $M^{2n+1}$ is equivalent to the assignment of G-structure G on M with the structural group $G=U\left(n\right)\times \left\{1\right\}$. The elements of the total space of this G-structure are complex frames of the M manifold of the form $p=\left(p,{\xi }_p,{ϵ}_1,\dots ,{ϵ}_n,{ϵ}_{\widehat{1}},\dots ,{ϵ}_{\widehat{n}}\right)$. These frames are characterized by the fact that the matrices of tensors $\mathrm{\Phi }$ and g in them have, respectively, the form:

$$\begin{array}{cc}\hfill & \left(1\right)\left({\mathrm{\Phi }}_i^j\right)=\left(\begin{array}{ccc}0& 0& 0\\ 0& \sqrt{-1}{I}_n& 0\\ 0& 0& -\sqrt{-1}{I}_n\end{array}\right),\\ & \left(2\right)\left(g_{ij}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& 0& I_n\\ 0& I_n& 0\end{array}\right),\end{array}$$

(4)

where $I_n$ is an identity matrix of order n. Let us assume that indexes i, j, k, …run through values from 0 to $2n$, and indexes a, b, c, d, …run values from 1 to n. Assume that $\widehat{a}=a+n$. It is well known [15] that the first group of structural equations of the G-structure G is the following:

$$\begin{array}{cc}\hfill & \left(1\right)d{\theta }^a=-{\theta }_b^a\wedge {\theta }^b+C^{ab}{}_c{\theta }^c\wedge {\theta }_b+C^{abc}{\theta }_b\wedge {\theta }_c+C^a{}_b{\theta }^b\wedge \theta +C^{ab}{\theta }_b\wedge \theta ;\\ & \left(2\right)d{\theta }_a={\theta }_a^b\wedge {\theta }_b+C_{ab}{}^c{\theta }_c\wedge {\theta }^b+C_{abc}{\theta }^b\wedge {\theta }^c+C_a{}^b{\theta }_b\wedge \theta +C_{ab}{\theta }^b\wedge \theta ;\\ & \left(3\right)d\theta =D_{ab}{\theta }^a\wedge {\theta }^b+D^{ab}{\theta }_a\wedge {\theta }_b+D_a^b{\theta }^a\wedge {\theta }_b+D_a\theta \wedge {\theta }^a+D^a\theta \wedge {\theta }_a,\end{array}$$

(5)

where $\left\{{\theta }_j^i\right\}$ are the components of the Riemannian connection form ∇ of metrics g, $\theta ={\theta }^0={\pi }^{*}\eta$, $\pi$ is a natural projection of the G-structure total space to the manifold M,

$$\begin{array}{cc}\hfill & \left(1\right){\mathrm{\Phi }}_{b,k}^a=0;\\ & \left(2\right){\mathrm{\Phi }}_{\widehat{b},k}^{\widehat{a}}=0;\\ & \left(3\right){\mathrm{\Phi }}_{0,k}^0=0;\\ & \left(4\right)C^{abc}=\frac{\sqrt{-1}}{2}{\mathrm{\Phi }}_{[\widehat{b},\widehat{c}]}^a;\\ & \left(5\right)C_{abc}=-\frac{\sqrt{-1}}{2}{\mathrm{\Phi }}_{[b,c]}^{\widehat{a}};\\ & \left(6\right)C^{ab}{}_c=-\frac{\sqrt{-1}}{2}{\mathrm{\Phi }}_{\widehat{b},c}^a;\\ & \left(7\right)C_{ab}{}^c=\frac{\sqrt{-1}}{2}{\mathrm{\Phi }}_{b,\widehat{c}}^{\widehat{a}};\\ & \left(8\right)C^{ab}=\sqrt{-1}(\frac{1}{2}{\mathrm{\Phi }}_{\widehat{b},0}^a-{\mathrm{\Phi }}_{0,\widehat{b}}^a);\\ & \left(9\right)C_{ab}=-\sqrt{-1}(\frac{1}{2}{\mathrm{\Phi }}_{b,0}^{\widehat{a}}-{\mathrm{\Phi }}_{0,b}^{\widehat{a}});\\ & \left(10\right)C^a{}_b=-\sqrt{-1}{\mathrm{\Phi }}_{0,b}^a;C_a{}^b=\sqrt{-1}{\mathrm{\Phi }}_{0,\widehat{b}}^{\widehat{a}};\\ & \left(11\right)D^{ab}=\sqrt{-1}{\mathrm{\Phi }}_{[\widehat{a},\widehat{b}]}^0;& \left(12\right)D_{ab}=-\sqrt{-1}{\mathrm{\Phi }}_{[a,b]}^0;\hfill \\ & \left(13\right)D_a^b=-\sqrt{-1}({\mathrm{\Phi }}_{a,\widehat{b}}^0+{\mathrm{\Phi }}_{\widehat{b},a}^0);\\ & \left(14\right)D^a=-\sqrt{-1}{\mathrm{\Phi }}_{\widehat{a},0}^0;\\ & \left(15\right)D_a=\sqrt{-1}{\mathrm{\Phi }}_{a,0}^0.\end{array}$$

(6)

Herewith

$$\begin{array}{cc}& \left(1\right)C^{abc}=-C^{acb};\\ & \left(2\right)C_{abc}=-C_{acb};\\ & \left(3\right)C^{ab}{}_c=-C^{ba}{}_c;\\ & \left(4\right)C_{ab}{}^c=C_{ba}{}^c;\\ & \left(5\right)D^{ab}=-D^{ba};\\ & \left(6\right)D_{ab}=-D_{ba};\\ & \left(7\right)D_a^b=C_a{}^b-C^b{}_a.\end{array}$$

(7)

Let us also recall that an almost Hermitian structure (in short $AH$-) a structure on the manifold M is called a pair $\left(J,g=⟨·,·⟩\right)$ of tensor fields on M, where J is an almost complex structure, $J^2=-id$, and g is a Riemannian metric. Herewith

$$⟨JX,JY⟩=⟨X,Y⟩;X,Y\in \mathcal{X}\left(M\right).$$

(8)

Setting an $AH$-structure to $M^{2n}$ is equivalent to plotting a G-structure on M with a structural group $U\left(n\right)$. The elements of the total space of this G-structure are complex frames of the manifold M, characterized by the fact that the matrices of tensors J and g in them have, respectively, the form:

$$\begin{array}{cc}& \left(1\right)\left(J_i^j\right)=\left(\begin{array}{cc}\sqrt{-1}{I}_n& 0\\ 0& -\sqrt{-1}{I}_n\end{array}\right),\\ & \left(2\right)\left(g_{ij}\right)=\left(\begin{array}{cc}0& I_n\\ I_n& 0\end{array}\right).\end{array}$$

(9)

It is well-known that the first group of structural equations of this G-structure is

$$\begin{array}{cc}& \left(1\right)d{\omega }^a=-{\omega }_b^a\wedge {\omega }^b+B^{ab}{}_c{\omega }^c\wedge {\omega }_b+B^{abc}{\omega }_b\wedge {\omega }_c;\\ & \left(2\right)d{\omega }_a={\omega }_a^b\wedge {\omega }_b+B_{ab}{}^c{\omega }_c\wedge {\omega }^b+B_{abc}{\omega }^b\wedge {\omega }^c.\end{array}$$

(10)

Here, $\left\{{\omega }_j^i\right\}$ are the components of the Riemannian connection form of metric g, and $\left\{{\omega }^i\right\}$ are the displacement form components.

$$\begin{array}{cc}& \left(1\right)B^{abc}=\overline{B_{abc}};\\ & \left(2\right)B^{ab}{}_c=\overline{B_{ab}{}^c}\end{array}$$

(11)

are components of the so-called structural and virtual tensors in the space of the associated G-structure. Herewith

$$\begin{array}{cc}& \left(1\right)B^{abc}=-B^{acb};\\ & \left(2\right)B_{abc}=-B_{acb};\\ & \left(3\right)B^{ab}{}_c=-B^{ba}{}_c;\\ & \left(4\right)B_{ab}{}^c=-B_{ba}{}^c\end{array}$$

(12)

(for details, see, for example, [16]).

Recall [4] that $AC$-structures of the class $W_1\oplus W_4$ in Gray–Hervella classification (Vaisman-Gray structures) on the manifold $M^{2n}$ are determined by the identity

$${\nabla }_X\left(\mathrm{\Psi }\right)(X,Y)=-\frac{1}{2(n-1)}\left\{⟨X,X⟩\delta \mathrm{\Psi }\left(Y\right)-⟨X,Y⟩\delta \mathrm{\Psi }\left(X\right)-⟨JX,Y⟩\delta \mathrm{\Psi }\left(JX\right)\right\},$$

(13)

where $\mathrm{\Psi }(X,Y)=⟨X,JY⟩$ is a fundamental form of the $AH$-structure, and $\delta$ is the codifferentiation operator. By direct calculation, it is verified that this identity is equivalent to the following relations in the space of the associated G-structure:

$$\begin{array}{cc}& \left(1\right)B^{\left[abc\right]}=B^{abc};\\ & \left(2\right)B_{\left[abc\right]}=B_{abc};\\ & \left(3\right)B^{ab}{}_c={\beta }^{[a}{\delta }_c^{b]};\\ & \left(4\right)B_{ab}{}^c={\beta }_{[a}{\delta }_{b]}^c;\end{array}$$

(14)

where $\left\{{\beta }_i\right\}$ are functions in the space of the associated G-structure, which are components of the so-called Lee form (see Definition 2).

Definition 2

([13]). The Lee form of the almost Hermitian structure $\left(J,\tilde{g}\right)$ on the manifold $M^{2n+2}$ is called the form

$$\alpha =\frac{1}{n}\delta \mathrm{\Psi }\circ J,$$

(15)

where

$$\mathrm{\Psi }(X,Y)=\tilde{g}(X,JY)$$

(16)

is a fundamental form of the structure, δ is codifferentiation operator. Vector β, which is dual to the Lee form, is called the Lee vector. In this research, the Lee form of the AC-structure is defined as the Lee form of the linear expansion.

It is not difficult to check that, in the space of the associated G-structure $\mathcal{G}$, components of the Lee vector (or Lee form) are found by the formula

$${\beta }^{\alpha }=\frac{2}{n}{B}^{\alpha \gamma }{}_{\gamma }$$

(17)

or taking into considering [5],

$$\begin{array}{cc}& \left(1\right){\beta }^a=\frac{2}{n}{C}^{ah}{}_h+\frac{1}{n}{D}^a;\\ & \left(2\right){\beta }^0=-\frac{\sqrt{2}}{n}{C}^h{}_h;\\ & \left(3\right){\beta }_a=\frac{2}{n}{C}_{ah}{}^h+\frac{1}{n}{D}_a;\\ & \left(4\right){\beta }_0=-\frac{\sqrt{2}}{n}{C}_h{}^h.\end{array}$$

(18)

Definition 3

([13]). AC-structure is called a nearly trans-Sasakian (in short NTS-) structure if its linear expansion belongs to the class $W_1\oplus W_4$ of almost Hermitian structures in Gray–Hervella classification. An AC-manifold provided with an NTS-structure is called an NTS-manifold.

The first group of structural equations of the $NTS$-structure on the space of the associated G-structure has the form [13]:

$$\begin{array}{cc}\hfill & \left(1\right)d{\theta }^a=-{\theta }_b^a\wedge {\theta }^b+C^{abc}{\theta }_b\wedge {\theta }_c+\frac{{\beta }^0}{\sqrt{2}}\theta \wedge {\theta }^a;\\ & \left(2\right)d{\theta }_a={\theta }_a^b\wedge {\theta }_b+C_{abc}{\theta }^b\wedge {\theta }^c+\frac{{\beta }_0}{\sqrt{2}}\theta \wedge {\theta }_a;\\ & \left(3\right)d\theta =\frac{1}{\sqrt{2}}\left({\beta }^0-{\beta }_0\right){\delta }_a^b{\theta }^a\wedge {\theta }_b,\end{array}$$

(19)

where

$$\begin{array}{cc}\hfill & \left(1\right)C^a{}_b=-\sqrt{-1}{\mathrm{\Phi }}_{0,b}^a=-\frac{1}{\sqrt{2}}{\beta }^0{\delta }_a^b;\\ & \left(2\right)C_a{}^b=\sqrt{-1}{\mathrm{\Phi }}_{0,\widehat{b}}^{\widehat{a}}=-\frac{1}{\sqrt{2}}{\beta }_0{\delta }_a^b;\\ & \left(3\right)D_b^a=-\sqrt{-1}\left({\mathrm{\Phi }}_{a,\widehat{b}}^0+{\mathrm{\Phi }}_{\widehat{b},a}^0\right)=\frac{1}{\sqrt{2}}\left({\beta }^0-{\beta }_0\right){\delta }_b^a;\\ & \left(4\right)C^{abc}=\frac{\sqrt{-1}}{2}{\mathrm{\Phi }}_{\widehat{b},\widehat{c}}^a;\\ & \left(5\right)C_{abc}=-\frac{\sqrt{-1}}{2}{\mathrm{\Phi }}_{b,c}^{\widehat{a}};\\ & \left(6\right)C^{\left[abc\right]}=C^{abc};\\ & \left(7\right)C_{\left[abc\right]}=C_{abc}.\end{array}$$

(20)

Theorem 1

([5]). An NTS-structure with an open contact form is homothetic to the Sasakian structure.

Given Theorem 1, it is natural to accept

Definition 4

([13]). An NTS-structure with a closed contact form is called an eigen NTS-structure.

Assume that M is an $NTS$-manifold with a closed contact form, then

$$d\eta =0.$$

(21)

So, according to (19:3),

$${\beta }^0={\beta }_0.$$

(22)

In this case,

$$\begin{array}{cc}& {\pi }^{*}\alpha ={\beta }_i{\theta }^i={\beta }_0{\theta }^0+{\beta }^0{\theta }_0={\beta }^0\left({\theta }^0+{\theta }_0\right)\\ & =\sqrt{2}{\beta }^0\theta =\sqrt{2}{\beta }^0{\pi }^{*}\eta .\end{array}$$

(23)

So, $\exists {\tilde{\beta }}^0\in C^{\infty }\left(M\right)$ and ${\beta }^0={\pi }^{*}{\tilde{\beta }}^0$. Put

$$\chi =-\frac{1}{\sqrt{2}}{\tilde{\beta }}^0.$$

(24)

Then, since

$$\alpha =\sqrt{2}{\tilde{\beta }}^0\eta ,$$

(25)

then

$$\alpha =-2\chi \eta .$$

(26)

And the first group of structural equations will take the form:

$$\begin{array}{cc}& \left(1\right)d{\theta }^a=-{\theta }_b^a\wedge {\theta }^b+C^{abc}{\theta }_b\wedge {\theta }_c+\chi {\theta }^a\wedge \theta ;\\ & \left(2\right)d{\theta }_a={\theta }_a^b\wedge {\theta }_b+C_{abc}{\theta }^b\wedge {\theta }^c+\chi {\theta }_a\wedge \theta ;\\ & \left(3\right)d\theta =0.\end{array}$$

(27)

Theorem 2

([5]). An AC-structure with a closed contact form on the manifold M is an eigen NTS-structure if and only if the identity is true.

$${\nabla }_X\left(\mathrm{\Phi }\right)Y+{\nabla }_Y\left(\mathrm{\Phi }\right)X=\chi \left\{\eta \left(X\right)\mathrm{\Phi }Y+\eta \left(Y\right)\mathrm{\Phi }X\right\},X,Y\in \mathcal{X}\left(M\right).$$

(28)

Theorem 3.

The class of NTS-manifolds with a closed contact form and a closed Lee form coincides with the class of AC-manifolds, with a closed contact form locally conformal to closely cosymplectic manifolds.

Proof. 

Assume that $\sigma \in C^{\infty }\left(M\right)$. Let us perform a conformal transformation with a defining function $\sigma$ of the eigen $NTS$-structure:

$$\tilde{g}=e^{-2\sigma }g;\tilde{\eta }=e^{-\sigma }\eta ;\tilde{\xi }=e^{\sigma }\xi .$$

(29)

Let $\tilde{\nabla }$ be a Riemannian connection of the transformed structure. Then, as is well-known (see, e.g.,: [4]), tensor T of the affinity deformation from connection ∇ to connection $\tilde{\nabla }$ looks like

$$T(X,Y)=⟨X,Y⟩\zeta -d\sigma \left(X\right)Y-d\sigma \left(Y\right)X,X,Y\in \mathcal{X}\left(M\right),$$

(30)

where $\zeta =grad\sigma$.

Therefore,

$${\tilde{\nabla }}_{X}Y={\nabla }_{X}Y+⟨X,Y⟩\zeta -d\sigma \left(X\right)Y-d\sigma \left(Y\right)X.$$

(31)

So,

$$\begin{array}{cc}& {\tilde{\nabla }}_X\left(\mathrm{\Phi }\right)Y={\nabla }_X\left(\mathrm{\Phi }\right)Y-\mathrm{\Phi }\left({\tilde{\nabla }}_{X}Y\right)\\ & ={\nabla }_X\left(\mathrm{\Phi }Y\right)+⟨X,\mathrm{\Phi }Y⟩\zeta -d\sigma \left(X\right)\mathrm{\Phi }Y-d\sigma \left(\mathrm{\Phi }Y\right)X\\ & -\mathrm{\Phi }\left({\nabla }_{X}Y\right)-⟨X,Y⟩\mathrm{\Phi }\zeta +d\sigma \left(X\right)\mathrm{\Phi }Y+d\sigma \left(Y\right)\mathrm{\Phi }X\\ & ={\nabla }_X\left(\mathrm{\Phi }\right)Y+⟨X,\mathrm{\Phi }Y⟩\zeta -d\sigma \left(\mathrm{\Phi }Y\right)X-⟨X,Y⟩\mathrm{\Phi }\zeta +d\sigma \left(Y\right)\mathrm{\Phi }X,\end{array}$$

(32)

i.e.,

$${\tilde{\nabla }}_X\left(\mathrm{\Phi }\right)Y={\nabla }_X\left(\mathrm{\Phi }\right)Y+⟨X,\mathrm{\Phi }Y⟩\zeta -d\sigma \left(\mathrm{\Phi }Y\right)X-⟨X,Y⟩\mathrm{\Phi }\zeta +d\sigma \left(Y\right)\mathrm{\Phi }X.$$

(33)

In particular, taking into consideration (28)

$$\begin{array}{cc}\hfill & {\tilde{\nabla }}_X\left(\mathrm{\Phi }\right)X={\nabla }_X\left(\mathrm{\Phi }\right)X-d\sigma \left(\mathrm{\Phi }X\right)X-{∥X∥}^2\mathrm{\Phi }\zeta +d\sigma \left(X\right)\mathrm{\Phi }X\\ & =\chi \eta \left(X\right)\mathrm{\Phi }X-d\sigma \left(\mathrm{\Phi }X\right)X+d\sigma \left(X\right)\mathrm{\Phi }X-{∥X∥}^2\mathrm{\Phi }\zeta .\end{array}$$

(34)

In particular, if the $\sigma$ function can be selected so that

$$d\sigma =-\chi \eta ,$$

(35)

then, obviously,

$$\zeta =-\chi \xi ,$$

(36)

and, taking into consideration the axioms of an $AC$-structure

$${\tilde{\nabla }}_X\left(\mathrm{\Phi }\right)X=0.$$

(37)

Moreover, in this case, due to the closure of the contact form $\eta$,

$$d\tilde{\eta }=d\left(e^{-\sigma }\eta \right)=-e^{-\sigma }d\sigma \wedge \eta =e^{-\sigma }\chi \eta \wedge \eta =0,$$

(38)

i.e.,

$$d\tilde{\eta }=0$$

(39)

then the transformed structure is closely cosymplectic, and the manifold M is locally conformally the closely cosymplectic manifold.

Next, assume that M is an $AC$-manifold with a closed contact form $\eta$, conformal to the closely cosymplectic manifold, and let $\sigma$ be the determining function of the corresponding conformal transformation of its $AC$-structure $\left(\xi ,\eta ,\mathrm{\Phi },g\right)\to \left(\tilde{\xi },\tilde{\eta },\mathrm{\Phi },\tilde{g}\right)$. Then,

$$\tilde{\eta }=e^{-\sigma }\eta ,$$

(40)

differentiating (40) externally, we obtain

$$0=d\tilde{\eta }=-e^{-\sigma }d\sigma \wedge \eta .$$

(41)

So, $d\sigma \wedge \eta =0$. Therefore, $\exists \chi \in C^{\infty }\left(M\right)$ and

$$d\sigma =-\chi \eta .$$

(42)

Accordingly, for the vector $\zeta$, dual to the $d\sigma$ form, we obtain

$$⟨\zeta ,X⟩=d\sigma \left(X\right)=-\chi \eta \left(X\right)=-\chi ⟨\xi ,X⟩,$$

(43)

and by virtue of the non-degeneracy of the metric,

$$\zeta =-\chi \xi .$$

(44)

Therefore, taking into consideration (34),

$$\begin{array}{cc}& 0={\tilde{\nabla }}_X\left(\mathrm{\Phi }\right)X\\ & ={\nabla }_X\left(\mathrm{\Phi }\right)X-d\sigma \left(\mathrm{\Phi }X\right)X-{∥X∥}^2\mathrm{\Phi }\zeta +d\sigma \left(X\right)\mathrm{\Phi }X\\ & ={\nabla }_X\left(\mathrm{\Phi }\right)X-\chi \eta \left(X\right)\mathrm{\Phi }X.\end{array}$$

(45)

Polarizing this identity, we obtain identity (28). By virtue of Theorem 2, the original structure is an eigen $NTS$-structure.

It remains to be noted that by virtue of Poincare lemma, the integrability condition of the Equation (35), i.e., the existence of a function $\sigma$, satisfying this equation, has the form

$$d\chi \wedge \eta =0,$$

(46)

which, by virtue of (26), is equivalent to the closure of the Lee form of the linear expansion of the $AC$-structure and, moreover, by virtue of (26), the function $\sigma$ is equal to half of the integral of the Lee form. □

Moreover, the defining function $\sigma$ of the locally-conformal transformation, found from Equation (35), in this case can also be found by the formula

$$\sigma =-ln\left(\chi \mu \right),$$

(47)

where $\mu$ is the integral of the contact form of the closely cosymplectic manifold.

An important example of eigen $NTS$-manifolds with a closed contact form and a closed form Lee are the Kenmotsu manifolds, i.e., the $AC$-manifolds characterized by the identity of

$${\nabla }_X\left(\mathrm{\Phi }\right)Y=⟨\mathrm{\Phi }X,Y⟩\xi -\eta \left(Y\right)\mathrm{\Phi }X;X,Y\in \mathcal{X}\left(M\right),$$

(48)

and the identity

$${\nabla }_X\left(\eta \right)Y=⟨X,Y⟩-\eta \left(X\right)\eta \left(Y\right);X,Y\in \mathcal{X}\left(M\right)$$

(49)

arising from it. This immediately follows from Theorem 2, if put in (28)

$$\chi =-1$$

(50)

and, therefore,

$$\alpha =2\eta ,$$

(51)

and by virtue of the latter identity

$$d\eta =0.$$

(52)

Kenmotsu manifolds were introduced in 1972 [17] and have a number of remarkable properties. Kenmotsu structures, for example, are defined on odd-dimensional Lobachevsky spaces of curvature −1. An exhaustive description of Kenmotsu structures is given in the paper [18]. The class of Kenmotsu manifolds coincides with the class of $AC$-manifolds obtained from cosymplectic manifolds by the canonical concircular transformation of the cosymplectic structure.

Another important example of eigen $NTS$-manifolds with a closed contact form and a closed form Lee are special generalized Kenmotsu manifolds of the second kind, i.e., the $AC$-manifold characterized by the identities

$$\begin{array}{cc}& \left(1\right){\nabla }_X\left(\mathrm{\Phi }\right)Y+{\nabla }_Y\left(\mathrm{\Phi }\right)X=-\eta \left(X\right)\mathrm{\Phi }Y+\eta \left(Y\right)\mathrm{\Phi }X;X,Y\in \mathcal{X}\left(M\right);\\ & \left(2\right){\nabla }_X\left(\eta \right)Y-{\nabla }_Y\left(\eta \right)X=0;X,Y\in \mathcal{X}\left(M\right).\end{array}$$

(53)

This immediately follows from (28), if we assume that

$$\chi =-1.$$

(54)

The local structure of these manifolds is as follows: the class of special generalized Kenmotsu manifolds of the second kind coincides with the class of $AC$-manifolds obtained from the closely cosymplectic manifolds by the canonical concircular transformation of the closely cosymplectic structure.

3. Harmonic NTS Manifolds

First, let us find out the geometric meaning of the function $\chi$ on the $NTS$-manifold M. To accomplish this, we calculate the codifferential of the contact form $\eta$. Since $\eta$ is a tensor of type (1, 0), then on the space of the bundle of frames over M we have:

$$d{\eta }_i+{\eta }_j{\theta }_i^j={\eta }_{i,j}{\theta }^j.$$

(55)

Because on space $\mathcal{G}$ of associated G-structure

$$\begin{array}{cc}& \left(1\right){\eta }_a={\eta }_{\widehat{a}}=0,\\ & \left(2\right){\eta }_0=1,\end{array}$$

(56)

narrowing of ratios (55) for $\mathcal{G}$ gives the following:

$$\begin{array}{cc}& \left(1\right){\theta }_a^0={\eta }_{a,j}{\theta }^j;\\ & \left(2\right){\eta }_{0,j}=0.\end{array}$$

(57)

But according to ([15] p. 74),

$${\theta }_a^0=\sqrt{-1}{\mathrm{\Phi }}_{a,j}^0{\theta }^j.$$

(58)

So,

$$\begin{array}{cc}& \left(1\right){\eta }_{a,j}=\sqrt{-1}{\mathrm{\Phi }}_{a,j}^0;\\ & \left(2\right){\eta }_{0,j}=0.\end{array}$$

(59)

However, according to (20),

$$\begin{array}{cc}& \left(1\right){\mathrm{\Phi }}_{a,\widehat{b}}^0=\sqrt{-1}{C}_a{}^b,\\ & \left(2\right){\mathrm{\Phi }}_{0,\widehat{b}}^{\widehat{a}}=-\sqrt{-1}{C}_a{}^b\end{array}$$

(60)

where

$${\eta }_{a,\widehat{b}}=-C_a{}^b=-\chi {\delta }_a^b.$$

(61)

Similarly,

$${\eta }_{\widehat{a},b}=-\chi {\delta }_b^a.$$

(62)

Further, according to (55) from [5],

$${\mathrm{\Phi }}_{a,b}^0+{\mathrm{\Phi }}_{b,a}^0=0.$$

(63)

On the other hand,

$$0=C_{ab}=-\sqrt{-1}{\mathrm{\Phi }}_{[a,b]}^0,$$

(64)

which means,

$${\mathrm{\Phi }}_{a,b}^0=0,$$

(65)

where

$${\eta }_{a,b}=0.$$

(66)

Thus, on the space of the associated G-structure, all components of the tensor $\nabla \eta$ are equal to zero, except

$$\begin{array}{cc}& \left(1\right){\eta }_{\widehat{a},b}=-\chi {\delta }_b^a,\\ & \left(2\right){\eta }_{b,\widehat{a}}=-\chi {\delta }_b^a,\end{array}$$

(67)

which is obviously equivalent to the identity

$${\nabla }_X\left(\eta \right)Y=\chi \left\{\eta \left(X\right)\eta \left(Y\right)-⟨X,Y⟩\right\};X,Y\in \mathcal{X}\left(M\right).$$

(68)

In particular,

$$\delta \eta \equiv g^{ij}{\eta }_{i,j}={\eta }_{}^a,a+{\eta }_{a,}{}^a=-2n\chi .$$

(69)

Therefore,

$$\chi =-\frac{1}{2n}\delta \eta .$$

(70)

Note that, due to the closure of the form $\eta$,

$$\Delta \eta \equiv (d\delta +\delta d)\eta =d\delta \eta =-2nd\chi ,$$

(71)

where $\Delta$ is a Laplace operator. In particular, the validity of the following theorem follows from this.

Theorem 4.

A contact form of a connected eigen NTS-manifold is harmonic if and only if $\chi =const$.

Definition 5

([13]). An NTS-manifold with a harmonic contact form is called the harmonic, and the number χ is its characteristic.

Example 1.

By virtue of Theorem 2 and the above, every Kenmotsu manifold is the harmonic $NTS$-manifold of the characteristic

$$\chi =-1.$$

(72)

Moreover, any special generalized Kenmotsu manifold of the second kind is the harmonic $NTS$-manifold of the characteristic

$$\chi =-1.$$

(73)

Example 2.

By virtue of Theorem 2, every closely cosymplectic manifold is the harmonic $NTS$-manifold characteristic

$$\chi =0.$$

(74)

What is more, it is fair.

Theorem 5.

Any compact harmonic NTS-manifold is the closely cosymplectic manifold with the unequal first Betty number.

Proof. 

Assume that $M^{2n+1}$ is a compact harmonic $NTS$-manifold with characteristics $\chi$. Then, by virtue of (70) and Green’s classical theorem,

$$-\frac{1}{2n}\chi VolM={\int }_M\left(-\frac{1}{2n}\chi \right)={\int }_M\delta \eta ={\int }_{M}div\xi =0,$$

(75)

so,

$$\chi =0.$$

(76)

But then, according to (28),

$${\nabla }_X\left(\mathrm{\Phi }\right)Y+{\nabla }_Y\left(\mathrm{\Phi }\right)X=0,$$

(77)

and $d\eta =0$.

So, M is the closely cosymplectic manifold. As

$$\eta \left(\xi \right)=1,$$

(78)

$\eta$ is non-zero harmonic form on a compact manifold M, then

$$dim{H}^1(M,R)\ne 0.$$

(79)

Note that in the case of the harmonic $NTS$-manifold M, it follows from (26) and Theorem 4 that the Lee form of such a manifold is automatically closed, and, by virtue of Theorem 3, this manifold is locally conformal to the closely cosymplectic manifold. Let us consider possible cases.

1. If $\chi =0$, then, according to (28),

$$\begin{array}{cc}& \left(1\right){\nabla }_X\left(\mathrm{\Phi }\right)Y+{\nabla }_Y\left(\mathrm{\Phi }\right)X=0,\\ & \left(2\right)d\eta =0,\end{array}$$

(80)

so, M is the closely cosymplectic manifold.

2. Assume that $\chi \ne 0$. Then, for the determining function $\sigma$, the conformal transformation of an $AC$-structure into the closely cosymplectic one, according to (42), we have

$$d\sigma =-\chi \eta ,$$

(81)

so,

$$\tilde{\eta }=e^{-\sigma }\eta =e^{-\sigma }\left(-\frac{1}{\chi }d\sigma \right)=d\left(\frac{1}{\chi }{e}^{-\sigma }\right).$$

(82)

Denote,

$$\frac{1}{\chi }{e}^{-\sigma }=\mu .$$

(83)

Then,

$$\sigma =-ln\left(\chi \mu \right).$$

(84)

Next, let $\left(\mathrm{\Phi },\tilde{\xi },\tilde{\eta },\tilde{g}\right)$ be a closely cosymplectic structure on the manifold M. Then

$$d\tilde{\eta }=0.$$

(85)

According to the Poincare lemma, in some neighborhood U of an arbitrary point $p\in M$ there is a smooth function $\mu$, such that

$$d\mu =\tilde{\eta }{|}_U.$$

(86)

Choosing a neighborhood U that is relatively compact and considering that the $\mu$ is defined to a constant, it can be considered that $\chi \mu >0$, where $\chi$ is a non-negative constant. Let us consider

$$\sigma =-ln\left(\chi \mu \right).$$

(87)

Now, perform a locally conformal transformation of the $AC$-structure $\left(\mathrm{\Phi },\tilde{\xi },\tilde{\eta },\tilde{g}\right)\stackrel{\beta }{\to }\left(\mathrm{\Phi },\xi ,\eta ,g\right)$, where

$$\begin{array}{cc}& \left(1\right)g=e^{2\sigma }\tilde{g};\\ & \left(2\right)\eta =e^{\sigma }\tilde{\eta };\\ & \left(3\right)\xi =e^{-\sigma }\tilde{\xi }.\end{array}$$

(88)

These ratios can be rewritten in the form:

$$\begin{array}{cc}\hfill & \left(1\right)g=\frac{1}{{\left(\chi \mu \right)}^2}\tilde{g};\\ & \left(2\right)\eta =\frac{1}{\chi \mu }\tilde{\eta };\\ & \left(3\right)\xi =\chi \mu \tilde{\xi }.\end{array}$$

(89)

If ∇ and $\tilde{\nabla }$ are the Levi-Civita connection of metrics g and $\tilde{g}$, then, as above, it is not difficult to calculate that a tensor T of affine deformation from connection $\tilde{\nabla }$ to ∇ looks like

$$T(X,Y)=d\sigma \left(X\right)Y+d\sigma \left(Y\right)X-\tilde{g}(X,Y)\zeta .$$

(90)

Note that, according to (84),

$$d\sigma =-\frac{1}{\mu }d\mu =-\frac{1}{\mu }\tilde{\eta },$$

(91)

hence, taking into considering (89),

$$T(X,Y)=-\frac{1}{\mu }\left\{\tilde{\eta }\left(X\right)Y+\tilde{\eta }\left(Y\right)X+\tilde{g}(X,Y)\tilde{\xi }\right\}.$$

(92)

Using this ratio as well as the ratio

$${\tilde{\nabla }}_X\left(\mathrm{\Phi }\right)X=0,$$

(93)

it is easy to calculate that

$$\begin{array}{cc}& {\nabla }_X\left(\mathrm{\Phi }\right)X=T(X,\mathrm{\Phi }X)-\mathrm{\Phi }\circ T(X,X)\\ & =-\frac{1}{\mu }\left\{\tilde{\eta }\left(X\right)\mathrm{\Phi }X+\tilde{\eta }\left(\mathrm{\Phi }X\right)+\tilde{g}(X,\mathrm{\Phi }X)\tilde{\xi }-2\tilde{\eta }\left(X\right)\mathrm{\Phi }X-\tilde{g}(X,X)\mathrm{\Phi }\tilde{\xi }\right\}\\ & =\frac{1}{\mu }\left\{\tilde{\eta }\left(X\right)\mathrm{\Phi }X\right\}=\chi \eta \left(X\right)\mathrm{\Phi }X.\end{array}$$

(94)

Additionally,

$$d\eta =d\left(e^{\sigma }\tilde{\eta }\right)=d\left(\frac{1}{\chi \mu }\tilde{\eta }\right)=-\frac{1}{\chi {\mu }^2}d\mu \wedge \tilde{\eta }=-\frac{1}{\chi {\mu }^2}\tilde{\eta }\wedge \tilde{\eta }=0,$$

(95)

this means that the transformed structure is the eigen $NTS$-structure. Let us call the constructed conformal transformation $\beta$ the closely cosymplectic structure as the canonical. Note

$$d\sigma =d(-ln\chi \mu )=-\frac{1}{\mu }d\mu =-\frac{1}{\mu }\tilde{\eta }=-\chi \eta .$$

(96)

This means that the conformal transformation $\alpha$ generated by the function $\sigma$ is the inverse of the transformation $\beta$. □

Note also that the identity (68) can be rewritten in the form

$$\nabla \eta =\chi (g-\eta \otimes \eta ),$$

(97)

and, by virtue of (35), in the form of

$$\nabla d\sigma ={\chi }^{2}g-d\sigma \otimes d\sigma .$$

(98)

This means that the function $\sigma$ satisfies the Yano equation [19], and the generated conformal transformation $\alpha$ is concircular, i.e., transfers geodesic circles into geodesic circles. Let us call it—the canonical concircular transformation of the harmonic $NTS$-structure. It is easy to see that the transformation inverse of the concircular is also concircular. In particular, the canonical transformation $\beta$ as an inverse conversion of $\alpha$ is also concircular. Thus, it is proved.

Theorem 6.

The class of harmonic NTS-manifolds coincides with the class of AC-manifolds obtained from the closely cosymplectic manifolds by canonical concircular transformation.

Since, as already mentioned, any closely cosymplectic manifold is locally equivalent to the Cartesian product of the nearly Kähler manifold and a real line [20], Theorem 6 gives a wide range of examples of harmonic $NTS$-manifolds: it is enough to take the Cartesian product of any nearly Kähler manifold M on the real line and produce a canonical concircular transformation of the closely cosymplectic structure of the manifold $M\times R$. The list of examples obtained will be exhaustive in the sense that, according to Theorem 6, any harmonic $NTS$-manifold is (locally) arranged in this way.

4. Conclusions

A wide class of harmonic nearly trans-Sasakian manifolds (i.e., nearly trans-Sasakian manifolds with harmonic contact form) has been identified and an exhaustive description of manifolds of this class has been obtained. It is proved that the class of harmonic $NTS$-manifolds coincides with the class of $AC$-manifolds obtained from closely cosymplectic manifolds by canonical concircular transformation. This article demonstrates examples of harmonic nearly trans-Sasakian manifolds. Since every closely cosymplectic manifold is locally equivalent to the product of a nearly Kähler manifold and a real line, it is sufficient to take the Cartesian product of any nearly Kähler manifold M and the real line to obtain examples of harmonic $NTS$-manifolds, as well as to produce a canonical concircular transformation of the closely cosymplectic structure of the $M\times R$ manifold. The list of obtained examples will be exhaustive in the sense that, according to Theorem 6, any harmonic $NTS$-manifold is (locally) arranged in this way.