Harmonic almost contact metric manifolds revisited

TL;DR

This paper extends the understanding of harmonic almost contact metric structures by generalizing previous characterizations and analyzing harmonicity as a map, with implications for their classification.

Contribution

It broadens the context for harmonicity conditions and explores harmonic maps and classification aspects of almost contact metric structures.

Findings

Harmonicity characterized in a more general setting.

Harmonicity as a map analyzed in this broader context.

Remarks provided on classification of structures.

Abstract

The study of harmonicity for almost contact metric structures was initiated by Vergara-D\'iaz and Wood and continued by Gonz\'alez-D\'avila and the present author. By using the intrinsic torsion and some restriction on the type of almost contact metric structure, Gonz\'alez-D\'avila and the present author have characterised harmonic structures by showing conditions relating harmonicity and classes of almost contact metric structures. Here we do this in a more general context. Moreover, the harmonicity of almost contact metric structures as a map is also done in such a general context. Finally, some remarks on the classification of almost contact metric structures are exposed.

Equations

$$⟨\Psi ,\Phi ⟩={\Psi }_{j_1\dots j_s}^{i_1\dots i_r}{\Phi }_{j_1\dots j_s}^{i_1\dots i_r},$$

$$⟨A,B{⟩}_{\mathcal{S}\mathcal{O}(M)/G}=⟨{\pi }_{\ast }A,{\pi }_{\ast }B⟩+⟨\varphi (A),\varphi (B)⟩.$$

$$\mathcal{E}(\sigma )=\frac{1}{2}{\int }_M\Vert {\sigma }_{\ast }{\Vert }^{2}dv,$$

$$d{\mathcal{E}}_{\sigma }(\phi )=-{\int }_M⟨{\xi }^G,\nabla \phi ⟩dv=-{\int }_M⟨d^{\ast }{\xi }^G,\phi ⟩dv.$$

$$(\mathrm{Hess}\mathcal{E}{)}_{\sigma }\phi$$

$${\nabla }^{\ast }\nabla \Psi ={\left({\nabla }_{e_i}^G{\xi }^G\right)}_{e_i}\Psi +{\xi }_{{\xi }_{e_i}^G{e}_i}^G\Psi -{\xi }_{e_i}^G({\xi }_{e_i}^G\Psi ).$$

$${\phi }^2=-I+\eta \otimes \zeta ,⟨\phi X,\phi Y⟩=⟨X,Y⟩-\eta (X)\eta (Y),$$

$$\text{mbox}{T}^{\ast }M\otimes u(n{)}^{\perp }=({\eta }^{\perp }\otimes u(n{)}_{|{\zeta }^{\perp }}^{\perp })\oplus (\eta \otimes u(n{)}_{|{\zeta }^{\perp }}^{\perp })\oplus ({\eta }^{\perp }\otimes {\eta }^{\perp }\wedge \eta )\oplus (\eta \otimes {\eta }^{\perp }\wedge \eta ).$$

$${\eta }^{\perp }\otimes u(n{)}_{|{\zeta }^{\perp }}^{\perp }={\mathcal{C}}_1\oplus {\mathcal{C}}_2\oplus {\mathcal{C}}_3\oplus {\mathcal{C}}_4,$$

$$\eta \otimes u(n{)}_{|{\zeta }^{\perp }}^{\perp }={\mathcal{C}}_{11},$$

$$\textstyle\otimes^{2}\eta^{\perp}=\mathbb{R}\langle\cdot,\cdot\rangle_{|\zeta^{\perp}}\oplus\operatorname{\mathfrak{su}}(n)_{s}\oplus\left\llbracket\sigma^{2,0}\right\rrbracket\oplus\mathbb{R}F\oplus\operatorname{\mathfrak{su}}(n)_{a}\oplus\operatorname{\mathfrak{u}}(n)^{\perp}_{|\zeta^{\perp}}.$$

$$\mathcal{C}_{5}\cong\mathbb{R}\langle\cdot,\cdot\rangle_{|\zeta^{\perp}},\quad\mathcal{C}_{8}\cong\operatorname{\mathfrak{su}}(n)_{s},\quad\mathcal{C}_{9}\cong\left\llbracket\sigma^{2,0}\right\rrbracket,\quad\mathcal{C}_{6}\cong\mathbb{R}F,\quad\mathcal{C}_{7}\cong\operatorname{\mathfrak{su}}(n)_{a},\quad\mathcal{C}_{10}\cong\operatorname{\mathfrak{u}}(n)^{\perp}_{|\zeta^{\perp}}.$$

$$\phi {\xi }_X^{\text{textsl}\mathrm{U}(n)}Y+{\xi }_X^{\text{textsl}\mathrm{U}(n)}\phi Y=\eta (Y)\phi {\xi }_X^{\text{textsl}\mathrm{U}(n)}\zeta +\eta ({\xi }_X^{\text{textsl}\mathrm{U}(n)}\phi Y)\zeta$$

$$\begin{gathered} {\xi }_X^{\text{textsl}\mathrm{U}(n)}=-\frac{1}{2}\phi \circ {\nabla }_X\phi +{\nabla }_X\eta \otimes \zeta -\frac{1}{2}\eta \otimes {\nabla }_X\zeta \\ =\frac{1}{2}({\nabla }_X\phi )\circ \phi +\frac{1}{2}{\nabla }_X\eta \otimes \zeta -\eta \otimes {\nabla }_X\zeta . \end{gathered}$$

$${\mathrm{Ric}}^{\ast }\in \mathbb{R}⟨\cdot ,\cdot ⟩\oplus \mathrm{su}(n{)}_s\oplus u(n{)}_{|{\zeta }^{\perp }}^{\perp }\oplus {\eta }_d^{\perp }\subseteq \mathbb{R}⟨\cdot ,\cdot ⟩\oplus \mathrm{su}(n{)}_s\oplus \eta \odot {\eta }^{\perp }\oplus u(n{)}_{|{\zeta }^{\perp }}^{\perp }\oplus \eta \wedge {\eta }^{\perp },$$

$${\mathrm{Ric}}_{\text{mbox}\mathrm{alt}}^{\ast }(X_{{\zeta }^{\perp }},Y_{{\zeta }^{\perp }})=$$

$$-({\xi }_{e_i}\eta )\wedge ({\xi }_{\phi e_i}\eta )\circ \phi (X_{{\zeta }^{\perp }},Y_{{\zeta }^{\perp }}),$$

$${\mathrm{Ric}}^{\ast }(\zeta )=$$

$$\tilde{\mathbf{a}}({\nabla }^{2}F{)}_{e_i,\phi e_i}(X,Y)=$$

$$-2⟨\phi {\xi }_{{\xi }_{e_i}\phi e_i}X,Y⟩+2⟨{\xi }_{{\xi }_{e_i}\phi e_i}\phi X,Y⟩$$

$$-2⟨{\xi }_{e_i}{\xi }_{\phi e_i}X,\phi Y⟩+2⟨{\xi }_{e_i}{\xi }_{\phi e_i}Y,\phi X⟩.$$

$${\mathrm{Ric}}_{\text{mbox}alt}^{\ast }(X,Y)=$$

$$+\eta \odot ({\xi }_{{\xi }_{e_i}\phi e_i}\eta )\circ \phi (X,Y)-({\xi }_{e_i}\eta )\wedge ({\xi }_{\phi e_i}\eta )\circ \phi (X,Y)$$

$$+\eta \wedge (\eta \circ {\xi }_{e_i}\circ {\xi }_{\phi e_i}\circ \phi )(X,Y).$$

$${\xi }_{(4)e_i}\phi e_i=-\frac{1}{2}(d^{\ast }F{)}^{♯}-\phi {\nabla }_{\zeta }\zeta +\frac{1}{2}{d}^{\ast }F(\zeta )\zeta ,{\xi }_{(6)e_i}\phi e_i=-d^{\ast }F(\zeta )\zeta .$$

$${\xi }_{(4)e_i}{e}_i=-\frac{1}{2}\phi (d^{\ast }F{)}^{♯}+\frac{1}{2}{\nabla }_{\zeta }\zeta ,{\xi }_{(5)e_i}{e}_i=-d^{\ast }\eta \zeta ,{\xi }_{(12)e_i}{e}_i=-{\nabla }_{\zeta }\zeta ,$$

$$4{\xi }_{(4)X}=X_{{\zeta }^{\perp }}^{♭}\otimes {\theta }^{♯}-\theta \otimes X_{{\zeta }^{\perp }}-\phi X^{♭}\otimes \phi {\theta }^{♯}+\phi \theta \otimes \phi X.$$

$$\phi {\xi }_{e_i}{\xi }_{\phi e_i}\zeta ={\xi }_{e_i}{\xi }_{e_i}\zeta -{\xi }_{(11)\zeta }{\xi }_{\zeta }\zeta +\Vert \nabla \zeta {\Vert }^2\zeta .$$

$$\phi {\xi }_{e_i}{\xi }_{\phi e_i}\zeta =-{\xi }_{e_i}{\xi }_{e_i}\zeta +{\xi }_{(11)\zeta }{\xi }_{\zeta }\zeta -\Vert \nabla \zeta {\Vert }^2\zeta .$$

$$\phi {\xi }_{e_i}{\xi }_{\phi e_i}\zeta =0,{\xi }_{e_i}{\xi }_{e_i}\zeta ={\xi }_{\zeta }{\xi }_{\zeta }\zeta ={\xi }_{(11)\zeta }{\xi }_{\zeta }\zeta -\Vert {\nabla }_{\zeta }\zeta {\Vert }^2\zeta \text{mbox}and\Vert \nabla \zeta \Vert =\Vert {\nabla }_{\zeta }\zeta \Vert .$$

$$\phi {\xi }_{e_i}{\xi }_{\phi e_i}\zeta$$

$${\sum }_{i=1}^{2n+1}{\xi }_{e_i}{\xi }_{e_i}\zeta ={\sum }_{i=1}^n{\xi }_{e_i}{\xi }_{e_i}\zeta +{\sum }_{i=1}^n{\xi }_{\phi e_i}{\xi }_{\phi e_i}\zeta +{\xi }_{\zeta }{\xi }_{\zeta }\zeta ={\xi }_{\zeta }{\xi }_{\zeta }\zeta .$$

$$\phi {\xi }_{e_i}{\xi }_{\phi e_i}\zeta =0,{\xi }_{e_i}{\xi }_{e_i}\zeta ={\xi }_{\zeta }{\xi }_{\zeta }\zeta ={\xi }_{(11)\zeta }{\xi }_{\zeta }\zeta -\Vert \nabla \zeta {\Vert }^2\zeta ,\text{mbox}and\Vert \nabla \zeta {\Vert }^2=\Vert {\nabla }_{\zeta }\zeta {\Vert }^2.$$

$${\eta }^{\perp }\cong [[{\lambda }^{1,0}]],u(n)\cong [{\lambda }^{1,1}],\mathrm{su}(n{)}_s\cong \mathrm{su}(n{)}_a\cong [{\lambda }_0^{1,1}],u(n{)}_{|{\zeta }^{\perp }}^{\perp }\cong [[{\lambda }^{2,0}]].$$

Full text

Keywords and phrases: $G$-structure, intrinsic torsion, minimal connection, almost contact metric structure, harmonic structure, harmonic unit vector field, harmonic map

2000 MSC: 53C10, 53C15, 53C25

1. Introduction

For an oriented Riemannian manifold $M$ of dimension $n$, given a Lie subgroup $G$ of $\operatorname{\textsl{SO}}(n)$, $M$ is said to be equipped with a $G$-structure, if there exists a subbundle $\mathcal{G}(M)$ of the oriented orthonormal frame bundle $\mathcal{SO}(M)$ with structural group $G$. For a fixed $G$, a natural question arises, ’which are the best $G$-structures on M?’. An approach to answer this question is based on the notion of the energy of a $G$-structure which is a particular case of the energy of a map between Riemannian manifolds. Such a functional has been widely studied by diverse authors [5, 6, 16]. The corresponding critical points are called harmonic maps and have been characterised by Eells and Sampson [7].

For principal $G$-bundles $Q\to M$ over a Riemannian manifold, Wood in [21] considers global sections $\sigma:M\to Q/H$ of the quotient bundle $\pi:Q/H\to M$, where $H$ is a Lie subgroup of $G$ such that $G/H$ is reductive. Such sections are in one-to-one correspondence with the $H$-reductions of the $G$-bundle $Q\to M$. Likewise, a connection on $Q\to M$ and a $G$-invariant metric on $G/H$ are fixed. Thus, $Q/H$ can be equipped in a natural way with a metric defined by using the metrics on $M$ and $G/H$. In such conditions, Wood regards harmonic sections as generalisations of harmonic maps from $M$ into $Q/H$, deriving the corresponding harmonic sections equations.

The situation described in the previous paragraph arises when the Riemannian manifold $M$ is equipped with some $G$-structure. Thus, in [9] it is considered $G$-structures defined on an oriented Riemannian manifold $M$ of dimension $n$, where $\operatorname{\textsl{G}}$ is a closed and connected subgroup of $\operatorname{\textsl{SO}}(n)$. Since the existence of a $G$-structure on $M$ is equivalent to the existence of a global section $\sigma:M\to\mathcal{SO}(M)/G$ of the quotient bundle, the energy of a $G$-structure is defined as the energy of the corresponding map $\sigma$. Such an energy functional is essentially determined by $\tfrac{1}{2}\int_{M}\|\xi^{G}\|^{2}\,dv$, where $\xi^{G}$ denotes the intrinsic torsion of the $G$-structure. As a consequence, the notion of harmonic $G$-structure, introduced by Wood in [21], is given in terms of the intrinsic torsion in [9]. This analysis has made possible to go further in the study of relations between harmonicity and classes of $G$-structures. Thus, the study of harmonic almost Hermitian structures initiated in [20, 21] is enriched in [9] with additional results.

Our purpose in the present work is going on the study of harmonicity for almost contact metric structures initiated by Vergara-D az and Wood in [18] and continued by Gonz lez-D vila and the present author in [11]. Almost contact metric structures can be seen as $\operatorname{\textsl{U}}(n)$-structures defined on manifolds of dimension $2n+1$. In [11], by using the intrinsic torsion and some restriction on the type of almost contact metric structure, type $\mathcal{C}_{1}\oplus\dots\oplus\mathcal{C}_{10}$, harmonic structures are characterised by showing conditions relating harmonicity and classes of almost contact metric structures. Here we do this for the general type $\mathcal{C}_{1}\oplus\dots\oplus\mathcal{C}_{12}$. Moreover, the harmonicity of the structure as a map is analyzed in such a general context.

In Section 5, some characterization of harmonic almost contact metric structures is firstly recalled in Theorem 5.1. Then we show conditions relating harmonicity and Chinea and Gonz lez-D vila’s classes [4] of almost contact metric structures. The harmonicity of such structures as a map of $M$ into $\mathcal{SO}(M)/(\operatorname{\textsl{U}}(n)\times 1)$ is also studied in Subsection 5.1. Section 5 ends by describing situations where the Reeb vector field is harmonic as unit vector field.

As a relevant remark, we point out the r le played by the identities given in Section 4. They are consequences of the trivial identities $d^{2}F=0$ and $d^{2}\eta=0$, where $F$ is the fundamental two-form of the almost contact metric structure and $\eta$ is the one-form metrically equivalent to the Reeb vector field $\zeta$ (see Section 3). Such identities are deduced by firstly expressing $d^{2}F$ and $d^{2}\eta$ in terms of the intrinsic torsion and the minimal connection, and then extracting certain $\operatorname{\textsl{U}}(n)$-components. Analog identities for almost Hermitian structures were deduced in [15, 14]. In the proofs of some theorems in Section 5, the use of these identities beside the harmonicity criteria is fundamental.

Finally, as another application of the identities in Section 4, we will derived some results relative to the classification of almost contact metric structures. In [13], results in such a direction have been already displayed. Here we derive another results with the same regards. In fact, the non-existence in a proper way of certain classes is proved.

2. Preliminaries

On an $n$-dimensional oriented Riemannian manifold $(M,\langle\cdot,\cdot\rangle),$ we consider the bundle $\pi_{\operatorname{\textsl{SO}}(n)}:\mathcal{SO}(M)\to M$ of the oriented orthonormal frames with respect to the metric $\langle\cdot,\cdot\rangle.$ Given a closed and connected subgroup $G$ of $\operatorname{\textsl{SO}}(n),$ a $G$-structure on $(M,\langle\cdot,\cdot\rangle)$ is a reduction $\mathcal{G}(M)\subset\mathcal{SO}(M)$ to $G.$ In the present Section we briefly recall some notions relative to $G$-structures (see [9, 11, 21] for more details).

Let ${\mathcal{S}\mathcal{O}}(M)/G$ be the orbit space under the action of $G$ on ${\mathcal{S}\mathcal{O}}(M)$ on the right. Then $\pi_{G}:{\mathcal{S}\mathcal{O}}(M)\to{\mathcal{S}\mathcal{O}}(M)/G$ is a principal $G$-bundle and \pi_{SO(n)}=\pi\makebox[7.0pt]{\raisebox{1.5pt}{\tiny\circ}}\pi_{G}, where $\pi:{\mathcal{S}\mathcal{O}}(M)/G\to M$ is a bundle with fibre $SO(n)/G$. The map $\sigma:M\to{\mathcal{S}\mathcal{O}}(M)/G$ given by $\sigma(m)=\pi_{G}(p),$ for all $p\in{\mathcal{G}}(M)$ with $\pi_{SO(n)}(p)=m,$ is a smooth section. Thus one has a one-to-one correspondence between the totally of $G$-structures and the manifold $\Gamma^{\infty}({\mathcal{S}\mathcal{O}}(M)/G)$ of all global sections of ${\mathcal{S}\mathcal{O}}(M)/G.$ Hence we will also denote by $\sigma$ the $G$-structure determined by a section $\sigma$.

The reduced subbundle $\mathcal{G}(M)$ gives rise to express the bundle of endomorphisms $\mbox{End}(\mbox{T}M)$ on the tangent bundle as the associated vector bundle $\mathcal{G}(M)\times_{G}\mbox{End}(\mathbb{R}^{n})$. We restrict our attention on the subbundle $\operatorname{\mathfrak{so}}(M)$ of $\mbox{End}(\mbox{T}M)$ of skew-symmetric endomorphisms $\varphi_{m}$, for all $m\in M$, i.e. $\langle\varphi_{m}X,Y\rangle=-\langle\varphi_{m}Y,X\rangle$. Note that this subbundle $\operatorname{\mathfrak{so}}(M)$ is expressed as $\operatorname{\mathfrak{so}}(M)=\mathcal{SO}(M)\times_{\operatorname{\textsl{SO}}(n)}\operatorname{\mathfrak{so}}(n)=\mathcal{G}(M)\times_{G}\operatorname{\mathfrak{so}}(n)$. Furthermore, because $\operatorname{\mathfrak{so}}(n)$ is decomposed into the $G$-modules $\operatorname{\mathfrak{g}},$ the Lie algebra of $G,$ and its orthogonal complement $\operatorname{\mathfrak{m}}$ with respect to the natural extension to endomorphisms of the Euclidean metric on $\mathbb{R}^{n}$, the bundle $\operatorname{\mathfrak{so}}(M)$ is also decomposed into $\operatorname{\mathfrak{so}}(M)=\operatorname{\mathfrak{g}}_{\sigma}\oplus\operatorname{\mathfrak{m}}_{\sigma}$, where $\operatorname{\mathfrak{g}}_{\sigma}=\mathcal{G}(M)\times_{G}\operatorname{\mathfrak{g}}$ and $\operatorname{\mathfrak{m}}_{\sigma}=\mathcal{G}(M)\times_{G}\operatorname{\mathfrak{m}}$.

Under the conditions above fixed, if $M$ is equipped with a $G$-structure, then there always exists a $G$-connection $\widetilde{\nabla}$ defined on $M$. Doing the difference $\widetilde{\xi}_{X}=\widetilde{\nabla}_{X}-\nabla_{X}$, where $\nabla_{X}$ is the Levi-Civita connection of $\langle\cdot,\cdot\rangle$, a tensor $\widetilde{\xi}_{X}\in\operatorname{\mathfrak{so}}(M)$ is obtained. Decomposing $\widetilde{\xi}_{X}=(\widetilde{\xi}_{X})_{\operatorname{\mathfrak{g}}_{\sigma}}+(\widetilde{\xi}_{X})_{\operatorname{\mathfrak{m}}_{\sigma}}$, $(\widetilde{\xi}_{X})_{\operatorname{\mathfrak{g}}_{\sigma}}\in\operatorname{\mathfrak{g}}_{\sigma}$ and $(\widetilde{\xi}_{X})_{\operatorname{\mathfrak{m}}_{\sigma}}\in\operatorname{\mathfrak{m}}_{\sigma}$, a new $G$-connection $\nabla^{G}$, defined by $\nabla^{G}_{X}=\widetilde{\nabla}_{X}-(\tilde{\xi}_{X})_{\operatorname{\mathfrak{g}}_{\sigma}}$, can be considered. Because the difference between two $G$-connections must be in $\operatorname{\mathfrak{g}}_{\sigma}$, $\nabla^{G}$ is the unique $G$-connection on $M$ such that its torsion $\xi^{G}_{X}=(\widetilde{\xi}_{X})_{\operatorname{\mathfrak{m}}_{\sigma}}=\nabla^{G}_{X}-\nabla_{X}$ is in $\operatorname{\mathfrak{m}}_{\sigma}$. $\nabla^{G}$ is called the minimal connection and $\xi^{G}$ is referred to as the intrinsic torsion of the $G$-structure $\sigma$ [8].

A natural way of classifying $G$-structures arises by decomposing the space $\mathcal{W}=\mbox{T}^{*}M\otimes\operatorname{\mathfrak{m}}_{\sigma}$ of possible intrinsic torsion into irreducible $G$-modules. This was initiated by Gray and Hervella [12] for almost Hermitian structures. In this particular case, $G=\operatorname{\textsl{U}}(n)$, the dimension of the manifold is $2n$ and the space $\mathcal{W}$ is decomposed into four irreducible $\operatorname{\textsl{U}}(n)$-modules. Therefore, $2^{4}=16$ classes of almost Hermitian structures were obtained.

Along the present paper, we will consider the natural extension of the metric $\langle\cdot,\cdot\rangle$ to $(r,s)$-tensors on $M$ defined by

[TABLE]

where the summation convention is used and $\Psi^{i_{1}\dots i_{r}}_{j_{1}\dots j_{s}}$ and $\Phi^{i_{1}\dots i_{r}}_{j_{1}\dots j_{s}}$ are the components of the $(r,s)$-tensors $\Psi,\Phi\in\mbox{T}_{s\,m}^{r}M$, with respect to an orthonormal frame over $m\in M$. Likewise, we will make reiterated use of the musical isomorphisms $\flat:\mbox{\rm T}M\to\mbox{\rm T}^{*}M$ and $\sharp:\mbox{\rm T}^{*}M\to\mbox{\rm T}M$, induced by the metric $\langle\cdot,\cdot\rangle$ on $M$, defined respectively by $X^{\flat}=\langle X,\cdot\rangle$ and $\langle\theta^{\sharp},\cdot\rangle=\theta$.

If $\omega$ is the connection one-form associated to $\nabla$, then $\mbox{T}\mathcal{SO}(M)=\ker\pi_{\operatorname{\textsl{SO}}(n)\ast}\oplus\ker\omega$. Now considering the projection $\pi_{G}\colon\mathcal{SO}(M)\to\mathcal{SO}(M)/G$, the tangent bundle of $\mathcal{SO}(M)/G$ is decomposed into $\mbox{T}\mathcal{SO}(M)/G=\mathcal{V}\oplus\mathcal{H}$, where $\mathcal{V}=\pi_{G\ast}(\ker\pi_{\operatorname{\textsl{SO}}(n)\ast})$ and $\mathcal{H}=\pi_{G\ast}(\ker\omega)$. For the projection $\pi:\mathcal{SO}(M)/G\to M$, $\pi(pG)=\pi_{\operatorname{\textsl{SO}}(n)}(p)$, the vertical and horizontal distributions $\mathcal{V}$ and $\mathcal{H}$ are such that $\pi_{\ast}\mathcal{V}=0$ and $\pi_{\ast}\mathcal{H}=\mbox{\rm T}M$. Moreover, it is considered the bundle $\pi^{*}\operatorname{\mathfrak{so}}(M)$ on $\mathcal{SO}(M)/G$ consisting of those pairs $(pG,\breve{\varphi}_{m})$, where $\pi(pG)=m$ and $\breve{\varphi}_{m}\in\operatorname{\mathfrak{so}}(M)_{m}$. Alternatively, $\pi^{*}\operatorname{\mathfrak{so}}(M)$ is described as the bundle $\pi^{*}\operatorname{\mathfrak{so}}(M)=\mathcal{SO}(M)\times_{G}\operatorname{\mathfrak{so}}(n)=\operatorname{\mathfrak{g}}_{\mathcal{SO}(M)}\oplus\operatorname{\mathfrak{m}}_{\mathcal{SO}(M)}$, where $\operatorname{\mathfrak{g}}_{\mathcal{SO}(M)}=\mathcal{SO}(M)\times_{G}\operatorname{\mathfrak{g}}$ and $\operatorname{\mathfrak{m}}_{\mathcal{SO}(M)}=\mathcal{SO}(M)\times_{G}\operatorname{\mathfrak{m}}$. A metric on each fibre in $\pi^{*}\operatorname{\mathfrak{so}}(M)$ is defined by $\langle(pG,\breve{\varphi}_{m}),(pG,\breve{\psi}_{m})\rangle=\langle\breve{\varphi}_{m},\breve{\psi}_{m}\rangle,$ where $\langle\cdot,\cdot\rangle$ in the right side is the metric on $(1,1)$-tensors on $M$ given by (2.1). With respect to this metric, the decomposition $\pi^{*}\operatorname{\mathfrak{so}}(M)=\operatorname{\mathfrak{g}}_{\mathcal{SO}(M)}\oplus\operatorname{\mathfrak{m}}_{\mathcal{SO}(M)}$ is orthogonal.

There is a canonical isomorphism between $\mathcal{V}$ and $\operatorname{\mathfrak{m}}_{\mathcal{SO}(M)}$. In fact, elements in $\operatorname{\mathfrak{m}}_{\mathcal{SO}(M)}$ can be seen as pairs $(pG,\breve{\varphi}_{m})$ such that if $\breve{\varphi}_{m}$ is expressed with respect to $p$, then it is obtained a matrix $(a_{ji})\in\operatorname{\mathfrak{m}}$. For all $a\in\operatorname{\mathfrak{m}}$, we have the fundamental vector field $a^{*}$ on $\mathcal{SO}(M)$ given by $a^{*}_{p}=\tfrac{d}{dt}_{|t=0}p.\exp ta\in\ker\pi_{\operatorname{\textsl{SO}}(n)*p}\subseteq\mbox{T}_{p}\mathcal{SO}(M).$ Any vector in $\mathcal{V}_{pG}$ is given by $\pi_{G*p}(a^{*}_{p})$, for some $a=(a_{ji})\in\operatorname{\mathfrak{m}}$. The isomorphism $\phi_{|\mathcal{V}_{pG}}\colon\mathcal{V}_{pG}\to\left(\operatorname{\mathfrak{m}}_{\mathcal{SO}(M)}\right)_{pG}$ is defined by $\phi_{|\mathcal{V}_{pG}}(\pi_{G*p}(a^{*}_{p}))=(pG,a_{ji}\,p(u_{i})^{\flat}\otimes p(u_{j})).$ Next it is extended the map $\phi_{|\mathcal{V}}:\mathcal{V}\to\operatorname{\mathfrak{m}}_{\mathcal{SO}(M)}$ to $\phi:\mbox{T}\,\mathcal{SO}(M)/G\to\operatorname{\mathfrak{m}}_{\mathcal{SO}(M)}$ by saying that $\phi(A)=0$, for all $A\in\mathcal{H}$, and $\phi(V)=\phi_{|\mathcal{V}}(V)$, for all $V\in\mathcal{V}$. This is used to define a metric $\langle\cdot,\cdot\rangle_{\mathcal{SO}(M)/G}$ on $\mathcal{SO}(M)/G$ by

[TABLE]

For this metric, $\pi\,:\,\mathcal{SO}(M)/G\to M$ is a Riemannian submersion with totally geodesic fibres (see [19] and [2, page 249]).

Now, we consider the set of all possible $G$-structures on a closed and oriented Riemannian manifold $M$ which are compatible with the metric $\langle\cdot,\cdot\rangle$. Such a set is identified with the manifold $\Gamma^{\infty}(\mathcal{SO}(M)/G)$ of all possible global sections $\sigma:M\to\mathcal{SO}(M)/G$. With respect to the metrics $\langle\cdot,\cdot\rangle$ and $\langle\cdot,\cdot\rangle_{\mathcal{SO}(M)/G}$, the energy of $\sigma$ is the integral

[TABLE]

where $\|\sigma_{\ast}\|^{2}$ is the norm of the differential $\sigma_{\ast}$ of $\sigma$ and $dv$ denotes the volume form on $(M,\langle\cdot,\cdot\rangle)$. On the domain of a local orthonormal frame field $\{e_{1},\dots,e_{n}\}$ on $M$, $\|\sigma_{*}\|^{2}$ can be locally expressed as $\|\sigma_{*}\|^{2}=\langle\sigma_{*}e_{i},\sigma_{*}e_{i}\rangle_{\mathcal{SO}(M)/G}$. Furthermore, using (2.2), from (2.3) it is obtained that the energy $\mathcal{E}(\sigma)$ of $\sigma$ is given by ${\mathcal{E}}(\sigma)=\frac{n}{2}{\rm Vol}(M)+\frac{1}{2}\int_{M}\|\phi\,\sigma_{*}\|^{2}dv.$ The relevant part of this formula $B(\sigma)=\frac{1}{2}\int_{M}\|\phi\,\sigma_{*}\|^{2}dv$ is called the total bending of the $G$-structure $\sigma$. In [9], it was shown that $\phi\,\sigma_{\ast}=-\xi^{G}$. Therefore, $B(\sigma)=\tfrac{1}{2}\int_{M}\|\xi^{G}\|^{2}\,dv.$

To study critical points of the functional $\mathcal{E}$ on $\Gamma^{\infty}(\mathcal{SO}(M)/G)$, smooth variations $\sigma_{t}\in\Gamma^{\infty}(\mathcal{SO}(M)/G)$ of $\sigma=\sigma_{0}$ are considered. The corresponding variation fields $m\to\varphi(m)=\frac{d}{dt}_{|t=0}\sigma_{t}(m)$ are sections of the induced bundle $\sigma^{*}\mathcal{V}$ on $M$. Furthermore, by using $\phi$, we will have $\phi\mbox{pr}_{2}^{\sigma}\sigma^{*}\mathcal{V}\cong\sigma^{*}\operatorname{\mathfrak{m}}_{\mathcal{SO}(M)}\cong\operatorname{\mathfrak{m}}_{\sigma}$. Thus, the tangent space $\mbox{T}_{\sigma}\Gamma^{\infty}(\mathcal{SO}(M)/G)$ is firstly identified with the space $\Gamma^{\infty}(\sigma^{*}\mathcal{V})$ of global sections of $\sigma^{*}\mathcal{V}$ [16]. A second identification is $\Gamma^{\infty}(\sigma^{*}\mathcal{V})\cong\Gamma^{\infty}(\operatorname{\mathfrak{m}}_{\sigma})$ as global sections of $\operatorname{\mathfrak{m}}_{\sigma}$.

In next theorem it is considered the coderivative $d^{*}\xi^{G}$ of the intrinsic torsion $\xi^{G}$, which is given by $d^{*}\xi^{G}_{m}=-(\nabla_{e_{i}}\xi^{G})_{e_{i}}=-(\nabla^{G}_{e_{i}}\xi^{G})_{e_{i}}-\xi^{G}_{\xi^{G}_{e_{i}}e_{i}}\in\operatorname{\mathfrak{m}}_{\sigma\,m},$ where $\{e_{1},\dots,e_{n}\}$ is any orthonormal frame on $m\in M$.

Theorem 2.1** ([9]).**

If $\,G$ is a closed and connected subgroup of $\operatorname{\textsl{SO}}(n)$, $(M,\langle\cdot,\cdot\rangle)$ a closed and oriented Riemannian manifold and $\sigma$ a global section of $\mathcal{SO}(M)/G$, then:

1. (i)

$($The first variation formula$)$. For the energy functional $\mathcal{E}:\Gamma^{\infty}(\mathcal{SO}(M)/G)\to\mathbb{R}$ and for all $\varphi\in\Gamma^{\infty}(\operatorname{\mathfrak{m}}_{\sigma})\cong\mbox{\rm T}_{\sigma}\Gamma^{\infty}(\mathcal{SO}(M)/G)$, we have

[TABLE] 2. (ii)

$($The second variation formula$)$. The Hessian form $({\rm Hess}\;{\mathcal{E}})_{\sigma}$ on $\Gamma^{\infty}(\operatorname{\mathfrak{m}}_{\sigma})$ is given by

[TABLE]

As a consequence of this Theorem the following notion is introduced: for general Riemannian manifolds $(M,\langle\cdot,\cdot\rangle)$ not necessarily closed and oriented, a $G$-structure $\sigma$ is said to be harmonic, if it satisfies $d^{*}\xi^{G}=0$ or, equivalently, $(\nabla^{G}_{e_{i}}\xi)_{e_{i}}=-\xi^{G}_{\xi^{G}_{e_{i}}e_{i}}$.

Given a $G$-structure $\sigma$ on a closed Riemannian manifold $(M,\langle\cdot,\cdot\rangle),$ the map $\sigma:(M,\langle\cdot,\cdot\rangle)\mapsto(\mathcal{SO}(M)/G,\langle\cdot,\cdot\rangle_{\mathcal{SO}(M)/G})$ is harmonic, i.e. $\sigma$ is a critical point for the energy functional on $\mathcal{C}^{\infty}(M,\mathcal{SO}(M)/G)$ if and only if its tension field $\tau(\sigma)=\left(\nabla_{e_{i}}\sigma_{*}\right)(e_{i})$ vanishes [16]. Here, $\nabla\sigma_{*}$ is defined by $\left(\nabla_{X}\sigma_{*}\right)(Y)=\nabla^{q}_{\sigma_{*}X}\sigma_{*}Y-\sigma_{*}(\nabla_{X}Y),$ where $\nabla^{q}$ denotes the induced connection by the Levi-Civita connection $\nabla^{q}$ of the metric in $\mathcal{SO}(M)/G$. According with [9, 21], harmonic sections $\sigma$ are characterised by the vanishing of the vertical component of $\tau(\sigma)$ and the horizontal component of $\tau(\sigma)$ is determined by the horizontal lift of the vector field metrically equivalent to the one-form $\nu_{\sigma},$ defined by $\nu_{\sigma}(X)=\langle\xi^{G}_{e_{i}},R_{e_{i},X}\rangle$. Hence one has the following

Proposition 2.2**.**

The map $\sigma:(M,\langle\cdot,\cdot\rangle)\mapsto(\mathcal{SO}(M)/G,\langle\cdot,\cdot\rangle_{\mathcal{SO}(M)/G})$ is a harmonic map if and only if $\sigma$ is a harmonic $G$-structure such that $\nu_{\sigma}=0$.

Relevant types of $G$-structures are those ones such that $\xi^{G}$ is metrically equivalent to a skew-symmetric three-form, i.e. $\xi^{G}_{X}Y=-\xi^{G}_{Y}X$. Next we recall some facts satisfied by them.

Proposition 2.3** ([9]).**

For a $G$-structure $\sigma$ such that $\xi^{G}_{X}Y=-\xi^{G}_{Y}X$, we have:

1. (i)

*If $[\xi^{G}_{X},\xi^{G}_{Y}]\subseteq\operatorname{\mathfrak{g}}_{\sigma}$, for all $X,Y\in\mathfrak{X}(M)$, then $\langle R_{X,Y\operatorname{\mathfrak{m}}_{\sigma}}X,Y\rangle=2\langle\xi^{G}_{X}Y,\xi^{G}_{X}Y\rangle$. * 2. (ii)

If $\sigma$ is a harmonic $G$-structure, then $\sigma$ is also a harmonic map.

In Section 5, we will study harmonicity of almost contact metric structures. Such structures are examples of $G$-structures defined by means of one or several $(r,s)$-tensor fields $\Psi$ which are stabilised under the action of $G$, i.e. $g\cdot\Psi=\Psi$, for all $g\in G$. Moreover, it will be possible to characterise the harmonicity of such $G$-structures by conditions given in terms of those tensors $\Psi$. The connection Laplacian (or rough Laplacian) $\nabla^{*}\nabla\Psi$ will play a relevant r le in such conditions. We recall that $\nabla^{*}\nabla\Psi=-\left(\nabla^{2}\Psi\right)_{e_{i},e_{i}},$ where $\{e_{1},\dots,e_{n}\}$ is an orthonormal frame field and $(\nabla^{2}\Psi)_{X,Y}=\nabla_{X}(\nabla_{Y}\Psi)-\nabla_{\nabla_{X}Y}\Psi$. If a Riemannian manifold $(M,\langle\cdot,\cdot\rangle)$ of dimension $n$ is equipped with a $G$-structure, where $G\subseteq\operatorname{\textsl{SO}}(n)$, and $\Psi$ is a $(r,s)$-tensor field on $M$ which is stabilised under the action of $G$, then

[TABLE]

As a consequence, if the $G$-structure is harmonic, then $\nabla^{*}\nabla\Psi=-\xi^{G}_{e_{i}}(\xi^{G}_{e_{i}}\Psi).$

3. Almost contact metric structures

An almost contact metric manifold is a $2n+1$-dimensional Riemannian manifold $(M,\langle\cdot,\cdot\rangle)$ equipped with a $(1,1)$-tensor field $\varphi$ and a unit vector field $\zeta$, called the Reeb vector field of the structure, such that

[TABLE]

where $\eta=\zeta^{\flat}$. Associated with such a structure the two-form $F=\langle\cdot,\varphi\cdot\rangle$, called the fundamental two-form, is usually considered. Using $F$ and $\eta$, $M$ can be oriented by fixing a constant multiple of $F^{n}\wedge\eta=F\wedge\stackrel{{\scriptstyle(n)}}{{\dots}}\wedge F\wedge\eta$ as volume form. Likewise, the presence of an almost contact metric structure is equivalent to say that $M$ is equipped with a $\operatorname{\textsl{U}}(n)\times 1$-structure. It is well known that $\operatorname{\textsl{U}}(n)\times 1$ is a closed and connected subgroup of $\operatorname{\textsl{SO}}(2n+1)$ and $\operatorname{\textsl{SO}}(2n+1)/(\operatorname{\textsl{U}}(n)\times 1)$ is reductive. In this case, the cotangent space on each point $\mbox{T}^{*}_{m}M$ is not irreducible under the action of the group $\operatorname{\textsl{U}}(n)\times 1$. In fact, $\mbox{T}^{*}M=\eta^{\perp}\oplus\mathbb{R}\eta$ and $\operatorname{\mathfrak{so}}(2n+1)\cong\Lambda^{2}\mbox{T}^{*}M=\Lambda^{2}\eta^{\perp}\oplus\eta^{\perp}\wedge\mathbb{R}\eta.$

From now on, we will denote $X_{\zeta^{\perp}}=X-\eta(X)\zeta$, for all $X\in\mathfrak{X}(M)$. Since $\Lambda^{2}\eta^{\perp}=\operatorname{\mathfrak{u}}(n)\oplus\operatorname{\mathfrak{u}}(n)^{\perp}_{|\zeta^{\perp}}$, where $\operatorname{\mathfrak{u}}(n)$ (resp., $\operatorname{\mathfrak{u}}(n)^{\perp}_{|\zeta^{\perp}}$) consists of those two-forms $b$ such that $b(\varphi X,\varphi Y)=b(X_{\zeta^{\perp}},Y_{\zeta^{\perp}})$ (resp., $b(\varphi X,\varphi Y)=-b(X_{\zeta^{\perp}},Y_{\zeta^{\perp}})$), we have $\operatorname{\mathfrak{so}}(2n+1)=\operatorname{\mathfrak{u}}(n)\oplus\operatorname{\mathfrak{u}}(n)^{\perp},\;$ with $\;\operatorname{\mathfrak{u}}(n)^{\perp}=\operatorname{\mathfrak{u}}(n)^{\perp}_{|\zeta^{\perp}}\oplus\eta^{\perp}\wedge\mathbb{R}\eta.$ Therefore, for the space $\mbox{T}^{*}M\otimes\operatorname{\mathfrak{u}}(n)^{\perp}$ of possible intrinsic $\operatorname{\textsl{U}}(n)\times 1$-torsions, we obtain

[TABLE]

In [4] it is showed that $\mbox{T}^{*}M\otimes\operatorname{\mathfrak{u}}(n)^{\perp}$ is decomposed into twelve irreducible $\operatorname{\textsl{U}}(n)$-modules $\mathcal{C}_{1},\dots,\mathcal{C}_{12}$, where

[TABLE]

The modules $\mathcal{C}_{1},\dots,\mathcal{C}_{4}$ are isomorphic to the Gray and Hervella’s $\operatorname{\textsl{U}}(n)$-modules above mentioned. Furthermore, note that $\varphi$ restricted to $\zeta^{\perp}$ works as an almost complex structure and, if one considers the $\operatorname{\textsl{U}}(n)$-action on the bilinear forms $\otimes^{2}\eta^{\perp}$, we have the decomposition

[TABLE]

The modules $\operatorname{\mathfrak{su}}(n)_{s}$ (resp., $\operatorname{\mathfrak{su}}(n)_{a}$) consists of Hermitian symmetric (resp., skew-symmetric) bilinear forms orthogonal to $\langle\cdot,\cdot\rangle_{|\zeta^{\perp}}$ (resp., $F$), and \left\llbracket\sigma^{2,0}\right\rrbracket (resp., $\operatorname{\mathfrak{u}}(n)^{\perp}_{|\zeta^{\perp}}$) is the space of anti-Hermitian symmetric (resp., skew-symmetric) bilinear forms. In relation with the modules $\mathcal{C}_{i}$, from $\eta^{\perp}\otimes\eta^{\perp}\wedge\mathbb{R}\eta\cong\otimes^{2}\eta^{\perp}$, using the $\operatorname{\textsl{U}}(n)$-map $\xi^{\operatorname{\textsl{U}}(n)}\to-\xi^{\operatorname{\textsl{U}}(n)}\eta=\nabla\eta$, it is obtained

[TABLE]

In summary, the space of possible intrinsic torsions $\mbox{T}^{*}M\otimes\operatorname{\mathfrak{u}}(n)^{\perp}$ consists of those tensors $\xi^{\operatorname{\textsl{U}}(n)}$ such that

[TABLE]

and, under the action of $U(n)\times 1$, is decomposed into:

1. (1)

if $n=1$, $\xi^{\operatorname{\textsl{U}}(1)}\in\mbox{T}^{*}M\otimes\operatorname{\mathfrak{u}}(1)^{\perp}=\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$; 2. (2)

if $n=2$, $\xi^{\operatorname{\textsl{U}}(2)}\in\mbox{T}^{*}M\otimes\operatorname{\mathfrak{u}}(2)^{\perp}=\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\dots\oplus\mathcal{C}_{12}$; 3. (3)

if $n\geqslant 3$, $\xi^{\operatorname{\textsl{U}}(n)}\in\mbox{T}^{*}M\otimes\operatorname{\mathfrak{u}}(n)^{\perp}=\mathcal{C}_{1}\oplus\dots\oplus\mathcal{C}_{12}$.

Now, we recall how some of these classes are referred to by diverse authors [3, 4]:

$\{\xi^{\operatorname{\textsl{U}}(n)}=0\}=$ cosymplectic manifolds, $\mathcal{C}_{1}=$ nearly-K-cosymplectic manifolds, $\mathcal{C}_{5}=$ $\alpha$-Kenmotsu manifolds, $\mathcal{C}_{6}=$ $\alpha$-Sasakian manifolds, $\mathcal{C}_{5}\oplus\mathcal{C}_{6}=$ trans-Sasakian manifolds, $\mathcal{C}_{2}\oplus\mathcal{C}_{9}=$ almost cosymplectic manifolds, $\mathcal{C}_{6}\oplus\mathcal{C}_{7}=$ quasi-Sasakian manifolds, $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}=$ quasi-K-cosymplectic manifolds, $\mathcal{C}_{3}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}=$ normal manifolds, $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}=$ almost $a$-Sasakian manifolds, etc.

The minimal $\operatorname{\textsl{U}}(n)$-connection is given by $\nabla^{\operatorname{\textsl{U}}(n)}=\nabla+\xi^{\operatorname{\textsl{U}}(n)}$ with

[TABLE]

For sake of simplicity, we will write $\xi=\xi^{\operatorname{\textsl{U}}(n)}$ in the sequel. Likewise, $\xi_{(i)}$ will denote the component of $\xi$ obtained by the $\operatorname{\textsl{U}}(n)$-isomorphism $(\nabla F)_{(i)}=(-\xi F)_{(i)}\in\mathcal{C}_{i}\to\xi_{(i)}$. In this way, classes or types are referred to as in [4].

Certain $\operatorname{\textsl{U}}(n)$-components of the Riemannian curvature tensor $R$ of an almost contact metric manifold are determined by a Ricci type tensor $\operatorname{Ric}^{*}$ associated to the structure, called the $\ast$-Ricci tensor. Such a tensor is defined by $\operatorname{Ric}^{*}(X,Y)=\langle R_{e_{i},X}\varphi e_{i},\varphi Y\rangle$. In general, $\operatorname{Ric}^{*}$ is not symmetric. However, since $\operatorname{Ric}^{*}$ satisfies the identities $\operatorname{Ric}^{*}(\varphi X,\varphi Y)=\operatorname{Ric}^{*}(Y_{\zeta^{\perp}},X_{\zeta^{\perp}})$, $\;\;\operatorname{Ric}^{*}(X,\zeta)=0,$ it can be claimed that

[TABLE]

where $\eta^{\perp}_{d}=\{2\eta\odot\alpha+\eta\wedge\alpha\,|\,\alpha\in\eta^{\perp}\}\cong\eta^{\perp}$ and we follow the convention $a\odot b=\tfrac{1}{2}(a\otimes b+b\otimes a)$.

The skew-symmetric part $\operatorname{Ric}^{*}_{\rm alt}$ of $\operatorname{Ric}^{*}$ will play a special rôle. Relative to $\operatorname{Ric}^{*}_{\rm alt}$, the following result was already given in [11]. However, there are some summands missing there.

Lemma 3.1**.**

Let $(M,\langle\cdot,\cdot\rangle,\varphi,\zeta)$ be a $2n+1$-dimensional almost contact metric manifold. Then the $*$-Ricci tensor satisfies

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$, and $\langle\operatorname{Ric}^{*}(\zeta),X\rangle=\operatorname{Ric}^{*}(\zeta,X)$. Furthermore, if $n>1$, we have:

1. (i)

The restriction $\operatorname{Ric}^{*}_{{\rm alt}|\zeta^{\perp}}$ of $\operatorname{Ric}^{*}_{\mbox{\rm\footnotesize alt}}$ to the space $\zeta^{\perp}$ is in $\operatorname{\mathfrak{u}}(n)^{\perp}_{|\zeta^{\perp}}$ and determines a $\operatorname{\textsl{U}}(n)$-component of the Weyl curvature tensor $W$. 2. (ii)

The vector field $\operatorname{Ric}^{*}(\zeta)$ is in $\zeta^{\perp}$ and determines another $\operatorname{\textsl{U}}(n)$-component of $W$.

*As a consequence, if $M$ is conformally flat and $n>1$, then $\operatorname{Ric}^{*}_{{\rm alt}|\zeta^{\perp}}=0$ and $\operatorname{Ric}^{*}(\zeta)=0$. *

Proof.

The proof follows in the same way as in [11]. However, we rewrite it because of missing summands in identities there. The so-called Ricci formula [2, p. 26] implies $-(R_{e_{i},\varphi e_{i}}F)(X,Y)=\tilde{\mathbf{a}}(\nabla^{2}F)_{e_{i},\varphi e_{i}}(X,Y),$ where $\tilde{\mathbf{a}}\colon T^{*}M\otimes T^{*}M\otimes\Lambda^{2}T^{*}M\to\Lambda^{2}T^{*}M\otimes\Lambda^{2}T^{*}M$ is the skewing mapping. On one hand, by making use of first Bianchi’s identity, it is straightforward to see $-(R_{e_{i},\varphi e_{i}}F)(X,Y)=4\operatorname{Ric}^{*}_{\mbox{\footnotesize alt}}(X,Y).$ On the other hand, it is relatively direct to check

[TABLE]

In [11] the last two summands are missing. Now, using equation (3.1), we will obtain the following right expression for $\operatorname{Ric}^{*}_{\mbox{\rm\footnotesize alt}}(X,Y)$:

[TABLE]

The tensors $\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi$ and $\xi$ are of the same type because $\nabla^{\operatorname{\textsl{U}}(n)}$ is a $\operatorname{\textsl{U}}(n)$-connection. Now, by replacing $X=X_{\zeta^{\perp}}$ and $Y=Y_{\zeta^{\perp}}$ (3.2), we will obtain the first required identity. Likewise, by replacing $X=\zeta$ and $Y=X$ in equation (3.2), the second required identity follows.

For the proof for the final assertions in the Lemma, see the one given in [11]. ∎

The vector field $\xi_{e_{i}}\varphi e_{i}$ involved in $\operatorname{Ric}^{*}$ is given by $\xi_{e_{i}}\varphi e_{i}=-\tfrac{1}{2}(d^{*}F)^{\sharp}-\tfrac{1}{2}d^{*}F(\zeta)\zeta-\varphi\nabla_{\zeta}\zeta.$ Thus, this vector field is contributed by the components of $\xi$ in $\mathcal{C}_{4}$ and $\mathcal{C}_{6}$. In fact,

[TABLE]

Likewise, the vector field $\xi_{e_{i}}e_{i}$ which is involved in the harmonicity criteria is given by $\xi_{e_{i}}e_{i}=-\tfrac{1}{2}\varphi(d^{*}F)^{\sharp}-d^{*}\eta\;\zeta-\tfrac{1}{2}\nabla_{\zeta}\zeta.$ Because

[TABLE]

one has that $\xi_{e_{i}}e_{i}$ is contributed by $\mathcal{C}_{4}$, $\mathcal{C}_{5}$ and $\mathcal{C}_{12}$.

For a $2n$-dimensional almost Hermitian manifold $(M,J,\langle\cdot,\cdot\rangle)$, where $J$ is the almost complex structure and $\langle\cdot,\cdot\rangle$ is the metric, the Lee one-form $\theta$ is defined by $\theta=-\frac{1}{n-1}Jd^{*}\omega$, where $\omega=\langle\cdot,J\cdot\rangle$ is the K hler two-form (see [12]). The one-form $\theta$ determines the component usually denoted by $\xi_{(4)}$ of the intrinsic torsion of the almost Hermitian structure. Such a component is given by $4\xi_{(4)X}=X^{\flat}\otimes\theta^{\sharp}-\theta\otimes X-JX^{\flat}\otimes J\theta^{\sharp}+J\theta\otimes JX.$ Note that $\sum_{i=1}^{2n}\xi_{e_{i}}e_{i}=\frac{n-1}{2}\theta^{\sharp}$.

In the context of almost contact metric geometric, taking (3.3) into account, the Lee form is defined by $(n-1)\theta=-\varphi(d^{*}F)^{\sharp}+\nabla_{\zeta}\eta,$ where $2n+1$ is the dimension of the almost contact metric manifold. The component $\xi_{(4)}$ is given by

[TABLE]

Likewise, for the vector field $\varphi\xi_{e_{i}}\xi_{\varphi e_{i}}\zeta$ involved in the expression for $\operatorname{Ric}^{*}(\zeta)$ obtained above, we have the results given in next lemma which will be useful later.

Lemma 3.2**.**

Denoting $\mathcal{A}=\mathcal{C}_{1}\oplus\mathcal{C}_{2}$, $\mathcal{B}=\mathcal{C}_{3}\oplus\mathcal{C}_{4}$, $\mathcal{C}=\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}$, $\mathcal{D}=\mathcal{C}_{9}\oplus\mathcal{C}_{10}$, $\mathcal{E}=\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ and being $\{e_{1},\dots,e_{2n},e_{2n+1}=\zeta\}$ an orthonormal basis for tangent vectors, we have:

1. $(i)$

If the almost contact metric structure is of type $\mathcal{A}\oplus\mathcal{B}\oplus\mathcal{C}\oplus\mathcal{E}$, then

[TABLE] 2. $(ii)$

If the almost contact metric structure is of type $\mathcal{A}\oplus\mathcal{B}\oplus\mathcal{D}\oplus\mathcal{E}$, then

[TABLE] 3. $(iii)$

If the almost contact metric structure is of type $\mathcal{A}\oplus\mathcal{C}\oplus\mathcal{E}$ or $\mathcal{B}\oplus\mathcal{D}\oplus\mathcal{E}$ or $\mathcal{C}_{1}\oplus\mathcal{C}\oplus\mathcal{C}_{9}\oplus\mathcal{E}$ or $\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{E}$ or $\mathcal{A}\oplus\mathcal{B}\oplus\mathcal{E}$, then

[TABLE]

Proof.

For (i) (for (ii)), by using (3.1), we have

[TABLE]

Note that the second summand of the right side is equal to zero and the required identity follows by considering the properties of the bilinear form $(\xi_{\cdot}\eta)\cdot$ acting on $\zeta^{\perp}$, i.e. in the case (i) (case (ii)), it is a Hermitian (skew Hermitian) bilinear form on $\zeta^{\perp}$. Finally, note that one has the identity $\xi_{\zeta}\xi_{\zeta}\zeta=-\|\nabla_{\zeta}\zeta\|^{2}\zeta+\xi_{(11)\zeta}\xi_{\zeta}\zeta.$

For the remaining particular cases, if the type is $\mathcal{A}\oplus\mathcal{C}\oplus\mathcal{E}$ or $\mathcal{B}\oplus\mathcal{D}\oplus\mathcal{E}$, we have $\sum_{i=1}^{n}\xi_{\varphi e_{i}}\xi_{\varphi e_{i}}\zeta=-\sum_{i=1}^{n}\xi_{e_{i}}\xi_{e_{i}}\zeta$. Therefore,

[TABLE]

For the type $\mathcal{A}\oplus\mathcal{B}\oplus\mathcal{E}$, as a consequence of $(i)$ and $(ii)$, it is followed

[TABLE]

The remaining cases are derived by a similar way using properties of the intrinsic torsion. ∎

Next it is pointed out more relative to notation.

Remark 3.3**.**

We will use the following standard notation: $\lambda_{0}^{p,q}$ is a complex irreducible $\operatorname{\textsl{U}}(n)$-module coming from the $(p,q)$-part of the complex exterior algebra, and that its corresponding dominant weight in standard coordinates is given by $(1,\dots,1,0,\dots,0,-1,\dots,-1)$, where $1$ and $-1$ are repeated $p$ and $q$ times, respectively. By analogy with the exterior algebra, there are also complex irreducible $\operatorname{\textsl{U}}(n)$-modules $\sigma^{p,q}_{0}$, with dominant weights $(p,0,\dots,0,-q)$ coming from the complex symmetric algebra. The notation $[\![V]\!]$ stands for the real vector space underlying a complex vector space $V$, and $[W]$ denotes a real vector space that admits $W$ as its complexification. Thus, being $A$ the irreducible $\operatorname{\textsl{U}}(n)$-module with dominant weight $(2,1,0,\dots,0)$, for the $\operatorname{\textsl{U}}(n)$-modules above mentioned one has

[TABLE]

The space of two forms $\Lambda^{2}\mathrm{T}^{*}M$ is decomposed into irreducible $\operatorname{\textsl{U}}(n)$-components as follows:

[TABLE]

The components of a two-form $\alpha$ are given by

[TABLE]

where $\lrcorner$ denotes the interior product and it is used the metric given by (2.1). In the sequel, we will consider the orthonormal basis for vectors $\{e_{1},\dots,e_{2n},e_{2n+1}=\zeta\}$. Likewise, we will use the summation convention. The repeated indexes will mean that the sum is extended from $i=1$ to $i=2n+1$. Otherwise, the sum will be explicitly written.

4. Geometric interrelations between components of the intrinsic torsion

In this section we will display several identities relating components of the intrinsic torsion of an almost contact streucture. Such identities were already obtained in [13] and are consequences of the equalities $d^{2}F=0$ and $d^{2}\eta=0$. They are interesting on their own and are applied in next section to derive conditions for the harmocinity of the structure. Likewise, the identities are also used to claim the non-existence of certain types of almost contact metric structures. This was already initiated in [13]. Here we will give further results in such a direction in the final section. The identity in next Lemma is a consequence of $d^{2}F=0$.

Lemma 4.1**.**

For almost contact metric manifolds of dimension $2n+1$, $n>1$, the following identity is satisfied

[TABLE]

In previous Lemma, if we use the equality

[TABLE]

we will obtain the $[\lambda^{1,1}]$-component of the exterior derivative of the Lee form $\theta$.

Proposition 4.2**.**

For almost contact metric manifolds of dimension $2n+1$, $n>1$, we have

[TABLE]

and $\textstyle(d\theta)_{\mathbb{R}}(X,Y)=\tfrac{1}{2n}\textstyle\langle d\theta,F\rangle F(X,Y),$ where

[TABLE]

The identity in next Lemma is also a consequence of $d^{2}F=0$.

Lemma 4.3** ([13]).**

For almost contact metric structures of dimension $2n+1$, $n>1$, the following identity is satisfied

[TABLE]

In previous Lemma, if we use the following identity proved in [9, p. 450, Lemma 4.4]

[TABLE]

we will obtain the $[\![\lambda^{2,0}]\!]$-component of the exterior derivative of the Lee form $\theta$.

Proposition 4.4**.**

For almost contact metric manifolds of dimension $2n+1$, $n>1$, the following identity is satisfied

[TABLE]

Next we give another consequence of $d^{2}F=0$.

Lemma 4.5** ([13]).**

For almost contact metric manifolds of dimension $2n+1$, the following identity is satisfied

[TABLE]

Next by noting that $(\nabla^{\operatorname{\textsl{U}}(n)}_{X}\xi_{(4)})_{e_{i}}e_{i}=\nabla^{\operatorname{\textsl{U}}(n)}_{X}\xi_{(4)e_{i}}e_{i}$ and using the identities

[TABLE]

another version of the identity in previous Lemma is given in next Proposition. Such a version relates the exterior derivatives of the Lee form $\theta$ and the coderivative $d^{*}\eta$.

Proposition 4.6**.**

For almost contact metric manifolds of dimension $2n+1$, we have

[TABLE]

In particular, if the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$ and $n>1$, then $d(d^{*}\eta)$ is given by $d(d^{*}\eta)=-d^{*}\eta\,\xi_{\zeta}\eta+d(d^{*}\eta)(\zeta)\eta$. Likewise, for the type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$ and $n>1$, the one-form $\mathrm{div}(\zeta)\;\eta=-d^{*}\eta\,\eta$ is closed.

If we consider the identity $d^{2}\eta=0$, we will obtain an expression for $d(d^{*}F(\zeta))$.

Lemma 4.7**.**

For almost contact metric manifolds of dimension $2n+1$, the exterior derivative $d(d^{*}F(\zeta))$ is given by

[TABLE]

[TABLE]

For our purposes, it is interesting to note that $(\nabla^{\operatorname{\textsl{U}}(n)}_{Z}\xi_{(6)})_{X}Y=\tfrac{1}{2n}d(d^{*}F(\zeta))(Z)(F(X,Y)\zeta+\eta(Y)\varphi(X)).$ From this, it is obtained $\textstyle((\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(6)})_{e_{i}}\eta)(Z)=-\frac{1}{2n}d(d^{*}F(\zeta))(\varphi Z).$

Also as a consequence of $d^{2}\eta=0$, one has the identities given in next Lemma already proved in [13, Lemma 3.9]. $(d\xi_{\zeta}\eta)_{V}$ will denote the projection of $d\xi_{\zeta}\eta$ on the $\operatorname{\textsl{U}}(n)$-space $V$.

Lemma 4.8**.**

The $\operatorname{\textsl{U}}(n)$-components of $d\xi_{\zeta}\eta$ are given by:

$(d\xi_{\zeta}\eta)_{\mathbb{R}\,F}=\frac{1}{2n}\langle d\xi_{\zeta}\eta,F\rangle\;F$, where

[TABLE]

[TABLE]

[TABLE]

$(d\xi_{\zeta}\eta)_{\eta\wedge[\![\lambda^{1,0}]\!]}=\eta\wedge\zeta\lrcorner d\xi_{\zeta}\eta$, where

[TABLE]

5. Harmonic almost contact structures

In this section we will show conditions relating harmonicity, curvature and types of almost contact metric structures. Firstly, we recall the following characterization for harmonic almost contact structures given in [11].

Theorem 5.1**.**

If $(M,\langle\cdot,\cdot\rangle,\varphi,\zeta)$ is a $2n+1$-dimensional almost contact metric manifold, then the following conditions are equivalent:

1. (i)

The structure is harmonic. 2. (ii)

For all $X,Y\in\mathfrak{X}(M)$, one has $\langle(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi)_{e_{i}}X_{\zeta^{\perp}},Y_{\zeta^{\perp}}\rangle+\langle\xi_{\xi_{e_{i}}e_{i}}X_{\zeta^{\perp}},Y_{\zeta^{\perp}}\rangle=0\,$ and $\,(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi)_{e_{i}}\zeta+\xi_{\xi_{e_{i}}e_{i}}\zeta=0$.

In such case we have $\nabla^{*}\nabla\zeta=-\xi_{e_{i}}\xi_{e_{i}}\zeta$. In particular, a structure of type $\mathcal{C}_{5}\oplus\ldots\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ is harmonic if and only if $\nabla^{*}\nabla\zeta=\|\nabla\zeta\|^{2}\zeta$, that is, the Reeb vector field $\zeta$ is harmonic unit vector field (see [17, 10] for this notion).

In next results, for certain types of almost contact metric structures, we will deduce conditions characterising harmonic structures. Such conditions are mainly given in terms of $\ast$-Ricci tensor and components of the intrinsic torsion.

Theorem 5.2**.**

For a $2n+1$-dimensional almost contact metric manifold $(M,\langle\cdot,\cdot\rangle,\varphi,\zeta)$, we have:

1. (i)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}=\mathcal{A}\oplus\mathcal{C}_{4}\oplus\mathcal{C}\oplus\mathcal{E}$, then the structure is harmonic if and only if

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$, and

[TABLE] 2. (ii)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\mathcal{\mathcal{missing}}C_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}=\mathcal{A}\oplus\mathcal{C}_{4}\oplus\mathcal{D}\oplus\mathcal{E}$, then the structure is harmonic if and only if

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$, and

[TABLE] 3. (iii)

If $M$ is of type $\mathcal{C}_{3}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}=\mathcal{B}\oplus\mathcal{C}\oplus\mathcal{E}$, then the structure is harmonic if and only if

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$, and

[TABLE]

Note that in this case of harmonic structure one has

[TABLE] 4. (iv)

If $M$ is of type $\mathcal{C}_{3}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}=\mathcal{B}\oplus\mathcal{D}\oplus\mathcal{E}$, then the structure is harmonic if and only if

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$, and

[TABLE]

Note that in this case of harmonic structure one has

[TABLE]

Proof.

For (i), the tensor $\xi$ for structures of type $\mathcal{A}\oplus\mathcal{C}_{4}\oplus\mathcal{C}\oplus\mathcal{E}$ is such that $\xi_{(\mathcal{A})\varphi X}\varphi Y=-\xi_{(\mathcal{A})X}Y$, $\xi_{(4)\varphi X}\varphi Y=\xi_{(4)X}Y$, and $(\xi_{\varphi X}\eta)(\varphi Y)=(\xi_{X_{\zeta^{\perp}}}\eta)(Y_{\zeta^{\perp}})$ [4]. Also in such a case, one has $\xi_{\xi_{(6)e_{i}}\varphi e_{i}}\zeta=-d^{*}F(\zeta)\xi_{\zeta}\zeta$ and $\xi_{\xi_{(4)e_{i}}\varphi e_{i}}\zeta=-\xi_{\varphi\xi_{(4)e_{i}}e_{i}}\zeta=-\tfrac{n-1}{2}\xi_{\varphi\theta^{\sharp}}\zeta$. Taking all of this into account, if the structure is harmonic, using Lemma 3.1, Theorem 5.1 and (4.4), it is followed the first condition required in (i).

The second condition in (i) is derived by making use of Lemma 3.1, Theorem 5.1, Lemma 3.2, Lemma 4.8 and the following identities

[TABLE]

Conversely, if both conditions are satisfied, using Lemma 3.1, Lemma 4.8, Lemma 3.2 and the identities (5.5)-(5.8), we will deduce the harmonicity conditions given by Theorem 5.1.

The proof for (ii) is similar. In this case one has $(\xi_{\varphi X}\eta)(\varphi Y)=-(\xi_{X_{\zeta^{\perp}}}\eta)(Y_{\zeta^{\perp}})$ and $\xi_{\xi_{e_{i}}\varphi e_{i}}\eta=\xi_{\xi_{(4)e_{i}}\varphi e_{i}}\zeta=-\xi_{\varphi\xi_{(4)e_{i}}e_{i}}\zeta=-\tfrac{n-1}{2}\xi_{\varphi\theta^{\sharp}}\zeta$.

For (iii), the intrinsic torsion in this case is such that $\xi_{(\mathcal{B})\varphi X}\varphi Y=\xi_{(\mathcal{B})X_{\zeta^{\perp}}}Y_{\zeta^{\perp}}$ and $(\xi_{\varphi X}\eta)(\varphi Y)=(\xi_{\zeta^{\perp}}\eta)(Y_{\zeta^{\perp}})$ (see [4]). Therefore, as in the previous cases, the required identities in (iii) are consequences of Lemma 3.1, Lemma 4.8, Theorem 5.1, Lemma 3.2 and identities (4.4), (5.6), (5.7), (5.8). Moreover, to deduce an expression for $\langle(\nabla^{\operatorname{\textsl{U}}(n)}_{\zeta}\xi_{(11)})_{\zeta}X,Y\rangle$, Lemma 4.3 is used. The converse is straightforward.

For (iv), $\xi$ in this case is such that $(\xi_{\varphi X}\eta)(\varphi Y)=-(\xi_{X_{\zeta^{\perp}}}\eta)(Y_{\zeta^{\perp}})$. Therefore, as in the previous cases, the required identities in (iv) are consequences of Lemma 3.1, Lemma 4.8, Theorem 5.1 and Lemma 3.2. Moreover, to deduce an expression for $\langle(\nabla^{\operatorname{\textsl{U}}(n)}_{\zeta}\xi_{(11)})_{\zeta}X,Y\rangle$, Lemma 4.3 is used. The converse easily follows by using results above mentioned. ∎

Next proposition contains some particular cases, most of them of previous Theorem. For proving them, identities of Section 4 are used.

Proposition 5.3**.**

For a $2n+1$-dimensional almost contact metric manifold $(M,\langle\cdot,\cdot\rangle,\varphi,\zeta)$, we have:*

1. (i)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$, and

[TABLE] 2. (ii)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$, and

[TABLE] 3. (iii)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$, then the structure is harmonic for $n\neq 5$ if and only if

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$, and it is satisfied the identity (5.2).

For $n=5$, the above mentioned type of structure is harmonic if and only if identities (5.1) and (5.2) are satisfied for this particular case. 4. (iv)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$, then the structure is harmonic for $n\neq 5$ if and only if

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$, and it is satisfied the identity (5.4).

For $n=5$, the above mentioned type of structure is harmonic if and only if identities (5.3) and (5.4) are satisfied for this particular case. 5. (v)

If $M$ is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}=\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\mathcal{C}\oplus\mathcal{E}$, then the structure is harmonic for $n\neq 2$ if and only if

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$, and it is satisfied the identity (5.2).

For $n=2$, the above mentioned type of structure is harmonic if and only if identities (5.1) and (5.2) are satisfied for this particular case. 6. (vi)

If $M$ is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}=\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\mathcal{D}\oplus\mathcal{E}$, then the structure is harmonic for $n\neq 2$ if and only if

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$, and it is satisfied the identity (5.4).

For $n=2$, the above mentioned type of structure is harmonic if and only if identities (5.3) and (5.4) are satisfied for this particular case. 7. (vii)

*If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if *

[TABLE]

for all $X,Y\in\mathfrak{X}(M)$, and it is satisfied the identity

[TABLE]

In particular, if the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}$, the structure is harmonic if and only if $\operatorname{Ric}^{*}(\zeta)=0$. Note that, for such a type, we always have $\operatorname{Ric}^{*}_{\rm alt}(X_{\zeta^{\perp}},Y_{\zeta^{\perp}})=0$, for all $X,Y\in\mathfrak{X}(M)$.

Proof.

Parts (i) and (ii) are merely particular cases of Theorem 5.2 (i) and (ii), respectively. For part (iii) and $n\neq 5$, from properties of the involved components of $\xi$, Proposition 4.4 and Theorem 5.1, we deduce that in this case

[TABLE]

[TABLE]

Then we take the first identity given in Lemma 3.1 and use properties of the involved components of $\xi$. In the resulting identity we replace $d\theta_{[\![\lambda^{2,0}]\!]}$ and $\langle(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(1)})_{e_{i}}X,Y\rangle$ by the expressions given above. The remaining second required identity and the particular case $n=5$ immediately follow from Theorem 5.2 (i).

For part (iv) and $n\neq 5$, from properties of the involved components of $\xi$, Proposition 4.4 and Theorem 5.1, we deduce that in this case

[TABLE]

[TABLE]

As before, we take the first identity given in Lemma 3.1 and use properties of the involved components of $\xi$. Then we replace $d\theta_{[\![\lambda^{2,0}]\!]}$ and $\langle(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(1)})_{e_{i}}X,Y\rangle$ by the expressions given above. The remaining second required identity and the particular case $n=5$ immediately follow from Theorem 5.2 (ii).

For part (v) and $n\neq 2$, from properties of the involved components of $\xi$, Proposition 4.4 and Theorem 5.1, we deduce that in this case

[TABLE]

[TABLE]

As in the previous cases, we take the first identity given in Lemma 3.1 and use properties of the involved components of $\xi$. Then, in the resulting identity, we replace $d\theta_{[\![\lambda^{2,0}]\!]}$ and $\langle(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(2)})_{e_{i}}X,Y\rangle$ by the expressions given above. The remaining second required identity and the particular case $n=2$ immediately follow from Theorem 5.2 (i).

For part (vi), from properties of the involved components of $\xi$, Proposition 4.4 and Theorem 5.1, we deduce that in this case $\tfrac{n-2}{2}d\theta_{[\![\lambda^{2,0}]\!]}(X,Y)=0,$ and

[TABLE]

As before, we take the first identity given in Lemma 3.1 and use properties of the involved components of $\xi$. Then we replace $d\theta_{[\![\lambda^{2,0}]\!]}$ and $\langle(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(2)})_{e_{i}}X,Y\rangle$ by the expressions given above. The remaining second required identity and the particular case $n=2$ immediately follow from Theorem 5.2 (ii).

For part (vii) and $n\neq 2$, from properties of the involved components of $\xi$, and Theorem 5.1, we deduce $\langle(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(1)})_{e_{i}}X,Y\rangle=0$ and the first required identity. The remaining second required identity follows from the second identity given in Lemma 3.1. Thus, due to the properties of the components of $\xi$, we firstly obtain

[TABLE]

Since, by one hand, one has

[TABLE]

and, by the other hand, it is satisfied

[TABLE]

by Theorem 5.1. Finally, using Lemma 4.8, we will obtain the second required identity. ∎

Next we focus attention on some particular cases where the harmonicity of the structure will require only one condition in all dimensions except for the dimension $5$, i.e. $n=2$. This is possible because it is used some of the identities obtained in Section 4. This is one of the applications of such identities. We write such cases as a corollary because they can be considered as consequences of previous Theorem and Proposition.

Corollary 5.4**.**

For a $2n+1$-dimensional almost contact metric manifold $(M,\langle\cdot,\cdot\rangle,\varphi,\zeta)$, we have:

1. (i)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if

[TABLE]

Note that, for such a type, $\operatorname{Ric}^{*}_{\mathrm{alt}}(X_{\zeta^{\perp}},Y_{\zeta^{\perp}})=\langle\xi_{(1)\xi_{\zeta}\zeta}X,Y\rangle$ and $(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(1)})_{e_{i}}=0$. 2. (ii)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if

[TABLE]

Note that, for such a type, $\operatorname{Ric}^{*}_{\mathrm{alt}}(X_{\zeta^{\perp}},Y_{\zeta^{\perp}})=\langle\xi_{(1)\xi_{\zeta}\zeta}X,Y\rangle$ and $(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(1)})_{e_{i}}=0$. 3. (iii)

If $M$ is of type $\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if

[TABLE]

Note that, for such a type, one has always $\operatorname{Ric}^{*}_{\mathrm{alt}}(X_{\zeta^{\perp}},Y_{\zeta^{\perp}})=(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(3)})_{e_{i}}X,Y\rangle=0$. 4. (iv)

If $M$ is of type $\mathcal{C}_{3}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if

[TABLE]

Note that, for such a type, one has always $\operatorname{Ric}^{*}_{\mathrm{alt}}(X_{\zeta^{\perp}},Y_{\zeta^{\perp}})=\langle(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(3)})_{e_{i}}X,Y\rangle=0$. 5. (v)

For $n\neq 2$, if $M$ is of type $\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if

[TABLE]

Note that, for such a type and dimension, $\operatorname{Ric}^{*}_{\mathrm{alt}}(X_{\zeta^{\perp}},Y_{\zeta^{\perp}})=\langle(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(4)})_{e_{i}}X,Y\rangle=d\theta_{[\![\lambda^{2,0}]\!]}(X,Y)=0.$ If $n=2$, the structure is harmonic if and only if it is satisfied (5.9) and $d\theta_{[\![\lambda^{2,0}]\!]}=0$. 6. (vi)

For $n\neq 2$, if $M$ is of type $\mathcal{C}_{4}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if

[TABLE]

Note that for such a type and dimension $\operatorname{Ric}^{*}_{\mathrm{alt}}(X_{\zeta^{\perp}},Y_{\zeta^{\perp}})=\langle(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(4)})_{e_{i}}X,Y\rangle=d\theta_{[\![\lambda^{2,0}]\!]}(X,Y)=0.$ If $n=2$, the structure is harmonic if and only if it is satisfied (5.10) and $d\theta_{[\![\lambda^{2,0}]\!]}=0$. 7. (vii)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if

[TABLE]

Note that, for such a type, $(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(1)})_{e_{i}}=0.$ 8. (viii)

If $M$ is of type $\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if it is satisfied (5.11).

Note that, for such a type, $(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(3)})_{e_{i}}=0.$ 9. (ix)

*For $n\neq 2$, if $M$ is of type $\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if *

[TABLE]

Note that, for such a type and dimension, $\langle(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(4)})_{e_{i}}X,Y\rangle=d\theta_{[\![\lambda^{2,0}]\!]}(X,Y)=0.$ If $n=2$, the structure is harmonic if and only if it is satisfied (5.12) and $d\theta_{[\![\lambda^{2,0}]\!]}=0$.

Proof.

For (i), (ii), (iii) and (iv), the first required identity by Theorem 5.1 is reduced to $(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(1)})_{e_{i}}=0$ in case of (i) and (ii) or $(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(3)})_{e_{i}}=0$ in case of (iii) and (iv), which follows from Proposition 4.4. The respective required condition is just the second condition of Theorem 5.2 (i), (ii), (iii) and (iv), respectively, applied to these particular cases.

For (v) and (vi), the first required identity by Theorem 5.1 is reduced to $d\theta_{[\![\lambda^{2,0}]\!]}=0$, which follows from Proposition 4.4 when $n\neq 2$. The respective required condition is just the second condition of Theorem 5.2 (iii) and (iv), respectively, applied to these cases.

The respective proofs for (vii), (viii) and (ix) are similar as for the previous cases but now we have to take also $\xi_{(9)}$ into account. The respective required identity it follows from the second identities in Theorem 5.1 and in Lemma 3.1. ∎

In the following corollaries some very special particular cases are displayed.

Corollary 5.5**.**

For a $2n+1$-dimensional almost contact metric manifold $(M,\langle\cdot,\cdot\rangle,\varphi,\zeta)$, we have:

1. (i)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}$ or $\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}$, the structure is harmonic if and only if $(dd^{*}\eta)_{|\zeta^{\perp}}=d(d^{*}F(\zeta))\circ\varphi$. In particular, structures of types $\mathcal{C}_{6}$, $\mathcal{C}_{1}\oplus\mathcal{C}_{5}$, $\mathcal{C}_{1}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}$ or $\mathcal{C}_{3}\oplus\mathcal{C}_{5}$ are harmonic. 2. (ii)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}$ or $\mathcal{C}_{3}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}$, the structure is harmonic if and only if $(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(9)})_{e_{i}}\zeta=0$. In particular, structures of type $\mathcal{C}_{1}\oplus\mathcal{C}_{10}$ or $\mathcal{C}_{3}\oplus\mathcal{C}_{10}$ are harmonic. 3. (iii)

For $n\neq 2$, if $M$ is of type $\mathcal{C}_{4}$, then the structure is harmonic. 4. (iv)

If $M$ is of type $\mathcal{C}_{12}$, the structure is harmonic if and only if the one-form $\xi_{\zeta}\eta$ is closed.

Proof.

For (i), the first identity of Theorem 5.1 is satisfied in this case because Proposition 4.4. The second required identity in such a Theorem is equivalent to $(dd^{*}\eta)_{|\zeta^{\perp}}-d(d^{*}F(\zeta))\circ\varphi=0$. This is due to the identities $(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(5)})_{e_{i}}\eta=\frac{1}{2n}(dd^{*}\eta)_{|\zeta^{\perp}}$, $(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(6)})_{e_{i}}\eta=-\frac{1}{2n}(d(d^{*}F(\zeta))\circ\varphi$, Proposition 4.6 and Lemma 4.7. The claims saying to be harmonic are deduced from the fact that $(dd^{*}\eta)_{|\zeta^{\perp}}=d(d^{*}F(\zeta))\circ\varphi=0$ in such cases.

For (ii), the first identity of Theorem 5.1 is satisfied by Proposition 4.4. The second required identity is equivalent to $(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi_{(9)})_{e_{i}}\zeta=0$. This is deduced from Proposition 4.6.

Parts (iii) and (iv) follow from Proposition 4.4 and Lemma 4.8, respectively. ∎

Corollary 5.6**.**

For a $2n+1$-dimensional almost contact metric manifold $(M,\langle\cdot,\cdot\rangle,\varphi,\zeta)$, we have:

1. (i)

If the structure is of type $\mathcal{C}_{5}$, $C_{6}$, $\mathcal{C}_{7}\oplus\mathcal{C}_{8}$, or $\mathcal{C}_{10}$, then the Reeb vector field $\zeta$ is harmonic unit vector field. 2. (ii)

For a conformally flat manifold $(M,\langle\cdot,\cdot\rangle)$ of dimension $2n+1$ with $n>1$, if an almost contact structure compatible with $\langle\cdot,\cdot\rangle$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}$, $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}$, $\mathcal{C}_{1}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}$ or $\mathcal{C}_{3}\oplus\mathcal{C}_{5}$, then it is harmonic.

Proof.

Part (i) follows from Corollary 5.6 and Theorem 5.1. Part (ii) is a consequence of the fact that, for those types, the harmonicity is equivalent to $\operatorname{Ric}^{*}_{\mathrm{alt}}=0$ and $\operatorname{Ric}^{*}(\zeta)=0$. Such conditions are satisfied due to the Weyl curvature vanishes (see Lemma 3.1). ∎

5.1. Harmonicity of almost

contact metric structures as a map

Now, we focus our attention on studying harmonicity as a map of almost contact metric structures, $\sigma:M\to\mathcal{SO}(M)/(\operatorname{\textsl{U}}(n)\times 1)$. Results in that direction were already obtained by Vergara-D az and Wood [18] and in [11] for the type $\mathcal{C}_{1}\oplus\dots\oplus\mathcal{C}_{10}$. We will complete such results for the general type $\mathcal{C}_{1}\oplus\dots\oplus\mathcal{C}_{12}$. In next Lemma, $s^{*}=\operatorname{Ric}^{*}(e_{i},e_{i})$ will denote the $*$-scalar curvature. If $\operatorname{Ric}^{*}(X,Y)=\frac{1}{2n}s^{*}(\langle X,Y\rangle-\eta(X)\eta(Y))$, the almost contact metric manifold is said to be weakly-ac-Einstein. If $s^{*}$ is constant, a weakly-ac-Einstein manifold is called ac-Einstein.

In Riemannian geometry, it is satisfied $2d^{*}\operatorname{Ric}+ds=0$, where $s$ is the scalar curvature. The $*$-analogue in almost contact metric geometry does not hold in general.

Lemma 5.7**.**

For almost contact metric manifolds of type $\mathcal{C}_{1}\oplus\ldots\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$, we have

[TABLE]

where $(\operatorname{Ric}^{*})^{t}(X,Y)=\operatorname{Ric}^{*}(Y,X)$ and $\xi_{X}^{\flat}(Y,Z)=\langle\xi_{X}Y,Z\rangle$. In particular, if the manifold is weakly-ac-Einstein, then

[TABLE]

Proof.

Note that $(\operatorname{Ric}^{*})^{t}(X,Y)=\tfrac{1}{2}\langle R_{e_{i},\varphi e_{i}}Y,\varphi X\rangle$. Then, we get

[TABLE]

[TABLE]

Since $\textstyle\sum_{j=1}^{2n}(\nabla_{e_{j}}\varphi)(e_{j})=2\varphi\xi_{(4)e_{j}}e_{j}+d^{*}F(\zeta)\zeta=(n-1)\varphi\theta^{\sharp}+d^{*}F(\zeta)\zeta$, by symmetry properties of $R$, it follows

[TABLE]

Using second Bianchi’s identity and taking into account

[TABLE]

we get

[TABLE]

Note that $\langle R_{X,e_{j}}e_{i},\nabla^{\operatorname{\textsl{U}}(n)}_{\varphi e_{j}}\varphi e_{i})\rangle=\langle R_{X,e_{j}}e_{i},e_{k}\rangle\langle\nabla^{\operatorname{\textsl{U}}(n)}_{\varphi e_{j}}\varphi e_{i},e_{k}\rangle=0,$ because it is a scalar product of a skew-symmetric matrix by a symmetric matrix. Finally, it is obtained

[TABLE]

Furthermore, ones obtains $ds^{*}(X)=\tfrac{1}{2}X\langle R_{e_{i},\varphi e_{i}}e_{j},\varphi e_{j}\rangle$. Hence

[TABLE]

But we also have that

[TABLE]

Thus,

[TABLE]

From (5.1) and (5.14), the required identity is obtained.

The claim relative to the case of weakly ac-Einstein follows by a straightforward way. Note that $2n\;(d^{*}\operatorname{Ric}^{*})^{t}=2n\;d^{*}\operatorname{Ric}^{*}=-ds^{*}-s^{*}\xi_{\zeta}\eta+(ds^{*}(\zeta)-s^{*}d^{*}\eta)\eta$ in such a case. ∎

Remark 5.8**.**

It is interesting to compare Lemma 5.7 with the analogous result for almost Hermitian geometry given in [9]. Thus, for an almost Hermitian manifold $(M,J,\langle\cdot,\cdot\rangle)$ of dimension $2n$, one has

[TABLE]

for all $X\in\mathfrak{X}(M)$. In particular, if the manifold is weakly $*$Einstein, then

[TABLE]

Next we focus our attention on conditions relative to harmonicity as a map of almost contact metric structures.

Theorem 5.9**.**

For an $2n+1$-dimensional almost contact metric manifold $(M,\langle\cdot,\cdot\rangle,\varphi,\zeta)$, we have:

1. (i)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$, then the structure is a harmonic map if and only if it is a harmonic structure and, for all $X\in\mathfrak{X}(M)$,

[TABLE] 2. (ii)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$, then the structure is a harmonic map if and only if it is a harmonic structure and, for all $X\in\mathfrak{X}(M)$,

[TABLE] 3. (iii)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}$, then the structure is a harmonic map if and only if $\operatorname{Ric}^{*}$ is symmetric and $(d^{*}\operatorname{Ric}^{*}+\tfrac{1}{2}ds^{*})(X)=3\langle R_{e_{i},X}\zeta,\xi_{e_{i}}\zeta\rangle$, for all $X\in\mathfrak{X}(M)$. Furthermore, one has the particular cases:

1. $(\mathrm{a})$

If the manifold is nearly-K-cosymplectic $(\mathcal{C}_{1})$, then structure is a harmonic map. 2. $(\mathrm{b})$

If the manifold is weakly-$ac$-Einstein, the structure is a harmonic map if and only if it is satisfied $(n-1)ds^{*}=4n\langle R_{e_{i},\cdot\,}\zeta,\xi_{e_{i}}\zeta\rangle+\left(\tfrac{n-1}{n}s^{*}d^{*}\eta+6\langle R_{e_{i},\zeta\,}\zeta,\xi_{e_{i}}\zeta\rangle\right)\eta.$ 4. (iv)

If $M$ is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}$, then the structure is a harmonic map if and only if it is a harmonic structure and, for all $X\in\mathfrak{X}(M)$,

[TABLE]

In particular, if the manifold is weakly-$ac$-Einstein, then the structure is a harmonic map if and only if it is a harmonic structure and satisfied $(n-1)ds^{*}=2n\langle R_{e_{i},\cdot\,}\zeta,\xi_{e_{i}}\zeta\rangle-2\langle R_{e_{i},\zeta\,}\zeta,\xi_{e_{i}}\zeta\rangle\,\eta.$ 5. (v)

If $M$ is of type $\mathcal{C}_{3}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$, then the structure is a harmonic map if and only if it is a harmonic structure and, for all $X\in\mathfrak{X}(M)$, satisfied

[TABLE] 6. (vi)

If $M$ is of type $\mathcal{C}_{3}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}$, then the structure is a harmonic map if and only it is a harmonic structure and, for all $X\in\mathfrak{X}(M)$, it is satisfied

[TABLE]

In particular, if the manifold is weakly-$ac$-Einstein, then the structure is a harmonic map if and only if the structure is harmonic and it is satisfied

[TABLE] 7. (vii)

If $M$ is of type $\mathcal{C}_{3}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$, then the structure is a harmonic map if and only if the structure is harmonic and, for all $X\in\mathfrak{X}(M)$,

[TABLE] 8. (viii)

If $M$ is of type $\mathcal{C}_{3}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}$, then the structure is a harmonic map if and only if it is a harmonic structure and, for all $X\in\mathfrak{X}(M)$,

[TABLE]

In particular:

1. (a)∗

If $\operatorname{Ric}^{*}$ is symmetric, then the structure is a harmonic map if and only if $\xi_{\theta^{\sharp}}=0$ and $d^{*}\operatorname{Ric}^{*}+\tfrac{1}{2}ds^{*}=-(n-1)\theta^{\sharp}\lrcorner\operatorname{Ric}^{*}-3\langle R_{e_{i},\cdot\;}\zeta,\xi_{e_{i}}\zeta\rangle.$ Furthermore, if the manifold is weakly-$ac$-Einstein, then the structure is a harmonic map if and only if $\xi_{\theta^{\sharp}}=0$ and $(n-1)ds^{*}=-(n-1)s^{*}\theta-6n\langle R_{e_{i},\,\cdot\,}\zeta,\xi_{e_{i}}\zeta\rangle+6\langle R_{e_{i},\,\zeta\,}\zeta,\xi_{e_{i}}\zeta\rangle\eta.$ 2. (b)∗

If the structure is of type $\mathcal{C}_{3}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}$, then the structure is a harmonic map if and only if $\operatorname{Ric}^{*}$ is symmetric and $d^{*}\operatorname{Ric}^{*}+\tfrac{1}{2}ds^{*}=-3\langle R_{e_{i},X}\zeta,\xi_{e_{i}}\zeta\rangle.$ Furthermore, if the manifold is also weakly-$ac$-Einstein, then the structure is a harmonic map if and only if $(n-1)ds^{*}=-6n\langle R_{e_{i},\,\cdot\,}\zeta,\xi_{e_{i}}\zeta\rangle+3\langle R_{e_{i},\,\zeta\,}\zeta,\xi_{e_{i}}\zeta\rangle\eta.$

Proof.

For (i), if the structure of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$, then by properties of the components of $\xi$ the identity in Lemma 5.7 can be expressed as

[TABLE]

for all $X\in\mathfrak{X}(M)$. From this identity and taking Proposition 2.2 into account, part (i) follows.

For (ii), in this case, if the structure of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$, the identity in Lemma 5.7 can be expressed as

[TABLE]

for all $X\in\mathfrak{X}(M)$. From this identity, using Proposition 2.2, part (ii) follows.

For (iii), it follows as particular case of (i) noting that some summands vanish. The particular case of structure of type $\mathcal{C}_{1}$ is derived from Corollary 5.5 (i) and Proposition 2.3. The another mentioned particular case, of weakly ac-Einstein, follows from the fact that $\operatorname{Ric}^{*}$ is symmetric in such a case and by using the second identity given in Lemma 5.7.

For (iv), we firstly note that if we have a harmonic structure of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}$, then $\operatorname{Ric}^{*}_{\mathrm{alt}}(X_{\zeta^{\perp}},Y_{\zeta^{\perp}})=0$. This implies $\langle\operatorname{Ric}^{*},\xi_{X}\rangle=\langle\operatorname{Ric}^{*}(\zeta),\xi_{X}\zeta\rangle$. Then applying the identity in (ii) to this particular case, it is deduced the required equality in (iv). The situation for weakly ac-Einstein manifolds is deduced by using the second identity given in Lemma 5.7.

The remaining parts (v), (vi),(vii) and (viii) are particular cases of the first four ones. They follow by using similar arguments as the previous ones. ∎

Remark 5.10**.**

See the analogous result in [9, Theorem 4.11, page 455].**

5.2. Harmonicity of the Reeb vector field as unit vector field

Now we analyze conditions for the harmonicity of the Reeb vector field as unit vector field in case of harmonic structure. From Theorem 5.1 and (2.4), the condition $\nabla^{*}\nabla\zeta=-\xi_{e_{i}}\xi_{e_{i}}\zeta$ is equivalent to the condition $\,(\nabla^{\operatorname{\textsl{U}}(n)}_{e_{i}}\xi)_{e_{i}}\zeta+\xi_{\xi_{e_{i}}e_{i}}\zeta=0$ which is the second condition to characterize the harmonicity of the structure. Thus, if the almost contact metric structure is of type $\mathcal{C}_{5}\oplus\dots\oplus\mathcal{C}_{12}$, then the structure is harmonic if and only if the Reeb vector field is harmonic unit vector field. In general, this equivalence is not true. However, in some cases, harmonicity of the structure implies harmonicity of the Reeb vector field. Results in this regard are the following ones.

Proposition 5.11**.**

If we have a manifold with harmonic almost contact metric structure of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{x}$ or $\mathcal{C}_{3}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{x}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{x}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{x}$ or $\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{x}$, where $x=11$ or $12$, then the Reeb vector field $\zeta$ is harmonic as unit vector field.

Proof.

It is an immediate consequence of Theorem 5.1. and Lemma 3.2. ∎

Example 5.12** **(The hyperbolic space).

The following example has been already considered in [4, 11, 13]. Let $\mathcal{H}=\{(x_{1},\dots,x_{2n+1})\in\mathbb{R}^{2n+1}\;|\;x_{1}>0\}$ be the $(2n+1)$-dimensional hyperbolic space with the Riemannian metric

[TABLE]

With respect to this metric, $\{E_{1},\ldots,E_{2n+1}\}$ is an orthonormal frame field, where $E_{i}=cx_{1}\frac{\partial}{\partial x_{i}}$, $i=1,\ldots,2n+1$. For the Lie brackets, one has $[E_{1},E_{j}]=cE_{j}$, $j=2,\ldots,2n+1$. The remaining Lie brackets relative to this frame are zero. The corresponding metrically equivalent coframe is $\{e_{1},\ldots,e_{2n+1}\}$, where $e_{i}=\tfrac{1}{cx_{1}}dx_{i}$. Note that $de_{i}=-ce_{1}\wedge e_{i}$, $i=1,\ldots,2n+1$.

The almost contact metric structure $(\varphi=\sum_{i,j=1}^{2n+1}\varphi^{i}_{j}e_{j}\otimes E_{i},\zeta,\eta,\langle\cdot,\cdot\rangle)$ is considered in [4]. The functions $\varphi^{i}_{j}$ are constant, $n\geq 2$ and $\zeta=\sum_{i=1}^{2n+1}x_{1}k_{i}\frac{\partial}{\partial x_{i}}=\sum_{i=1}^{2n+1}\frac{k_{i}}{c}E_{i}$, being $k_{i}=$ constant and $k_{1}^{2}+\ldots+k_{2n+1}^{2}=c^{2}$. The one-form $\eta$ and the fundamental form $F$ are given by $\textstyle\eta=\sum_{i=1}^{2n+1}\frac{k_{i}}{c}e_{i},\;\;F=\sum_{i,j=1}^{2n+1}\varphi^{i}_{j}e_{i}\wedge e_{j}.$ Their exterior derivatives are expressed as $d\eta=-ce_{1}\wedge\eta$, $dF=-2ce_{1}\wedge F.$ Hence $d\eta=\xi_{\zeta}\eta\wedge\eta\in[\![\lambda^{1,0}]\!]\cong\mathcal{C}_{12}$, where $\xi_{\zeta}\eta=k_{1}\eta-ce_{1}$, and $dF=2(k_{1}\eta-ce_{1})\wedge F-2k_{1}\eta\wedge F\in[\![\lambda^{1,0}]\!]\wedge F+\mathbb{R}\eta\wedge F\cong\mathcal{C}_{4}\oplus\mathcal{C}_{5}$. Since $dF_{[\![\lambda^{1,0}]\!]\wedge F}=\theta\wedge F$ and $dF_{\mathbb{R}\eta\wedge F}=-\frac{d^{*}\eta}{n}\eta\wedge F$, one has $\langle\cdot\lrcorner dF,F\rangle=2(n-1)\theta-2d^{*}\eta\,\eta$. Taking this into account we obtain

[TABLE]

Now, by using the Lie brackets described above, one can check that $N_{\varphi}(E_{i},E_{j})=0$. From all of this, for $n>1$, the structure is of type $\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{12}$ and, for $n=1$, it is of type $\mathcal{C}_{5}\oplus\mathcal{C}_{12}$.

Note that $(n-1)d\theta=(n-1)k_{1}\theta\wedge\eta$, $d\xi_{\zeta}\eta=k_{1}\xi_{\zeta}\eta\wedge\eta$ and $d(d^{*}\eta)=0$. From $d\xi_{\zeta}\eta=k_{1}\xi_{\zeta}\eta\wedge\eta$, using Lemma 4.8, we deduce $(\nabla^{\operatorname{\textsl{U}}(n)}_{\zeta}\xi)_{\zeta}\zeta=0$.

Particular cases are:

(i) $k_{1}=0$ and $n>1$. The structure is of strict type $\mathcal{C}_{4}\oplus\mathcal{C}_{12}$. The one-forms $\theta$ and $\xi_{\zeta}\eta$ are closed. In fact, $\xi_{\zeta}\eta=-d(\ln x_{1})$. If we do the conformal change of metric $x_{1}^{2}\langle\cdot,\cdot\rangle$, we obtain the flat cosymplectic structure on $\mathcal{H}$ as an open set of $\mathbb{R}^{2n+1}$ with the Euclidean metric.

(ii) $k_{1}\neq 0$, $k_{1}\neq c$ and $n=1$. The structure is of strict type $\mathcal{C}_{5}\oplus\mathcal{C}_{12}$.

(iii) $k_{1}=0$ and $n=1$. The structure is of strict type $\mathcal{C}_{12}$ and $\xi_{\zeta}\eta$ is closed.

(iv) $k_{1}=c$. The structure is of strict type $\mathcal{C}_{5}$ with $d^{*}\eta=2nc$.

Considering again the general case, since $\mathcal{H}$ has constant sectional curvarture, one has that it is ac-Einstein with $s^{*}=-2nc^{2}$. Therefore, $\operatorname{Ric}^{*}_{\mathrm{alt}}=0$ and $\operatorname{Ric}^{*}(\zeta)=0$ (note also that $d\theta_{[\![\lambda^{2,0}]\!]}=0$). For having a harmonic structure, by Corollary 5.4 (v), we need the condition $0=n\,k_{1}\,\xi_{\zeta}\eta.$ Therefore, we have:

The structure of type $\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{12}$, for $k_{1}\neq 0$, $k_{1}\neq c$ and $n>1$, is not harmonic. Applying in this case the identity (2.4), we obtain $\nabla^{*}\nabla\zeta=-(2n-1)k_{1}\,\xi_{\zeta}\zeta+((2n-1)k_{1}^{2}+c^{2})\zeta$. Hence the Reeb vector field $\zeta$ is not harmonic unit vector field. 2. -

The structure of type $\mathcal{C}_{4}\oplus\mathcal{C}_{12}$, for $k_{1}=0$ and $n>1$, is harmonic. However, when one tries to check the condition in Theorem 5.9 (v), one obtains $c^{2}\xi_{\zeta}\eta\neq 2nc^{2}\xi_{\zeta}\eta$. Hence the structure is not harmonic map. The Reeb vector field $\zeta$ is harmonic unit vector field. In fact, $\nabla^{*}\nabla\zeta=-\xi_{\zeta}\xi_{\zeta}\zeta=\|\xi_{\zeta}\zeta\|^{2}\zeta=c^{2}\zeta$, in according with Proposition 5.11. 3. -

The structure of type $\mathcal{C}_{5}\oplus\mathcal{C}_{12}$, for $n=1$, $k_{1}\neq 0$ and $k_{1}\neq c$, is not harmonic. Since $\nabla^{*}\nabla\zeta=-k_{1}\,\xi_{\zeta}\zeta+(k_{1}^{2}+c^{2})\zeta$, the Reeb vector field $\zeta$ is not harmonic unit vector field. 4. -

The structure of type $\mathcal{C}_{12}$, for $k_{1}=0$ and $n=1$, is harmonic. This was expected because the one-form $\xi_{\zeta}\eta$ is closed. Since $\sum_{i=1}^{3}\langle R_{e_{i},\cdot},\xi_{e_{i}}\rangle=\langle R_{\zeta,\cdot},\xi_{\zeta}\rangle=-2c^{2}\xi_{\zeta}\eta\neq 0$, the structure is not harmonic map. The Reeb vector field $\zeta$ is a harmonic unit vector field, i.e. $\nabla^{*}\nabla\xi=-\xi_{\zeta}\xi_{\zeta}\zeta=\|\xi_{\zeta}\zeta\|^{2}\zeta=c^{2}\zeta$. This is agree with Proposition 5.11. 5. -

The structure of type $\mathcal{C}_{5}$, for $k_{1}=c$, is harmonic. This was expected. For the condition in Theorem 5.9 (iii) (b), we have $(n-1)ds^{*}=0$ and the right side is equal to $4n(n+4)c^{3}\,\eta\neq 0$. Hence the structure is not harmonic map. Of course, we already know $\zeta$ must be harmonic unit vector field. In fact, $\nabla^{*}\nabla\zeta=-\xi_{e_{i}}\xi_{e_{i}}\zeta=2nc^{2}\,\zeta$.

6. Non-existence of certain types of almost contact metric structures

In this section we point out another application of the identities given in Section 4 as tools to prove the non-existence in a proper way of certain types of almost contact metric structures. Results in this direction have been derived in [13]. Here we do a further and more complete analysis.

Theorem 6.1**.**

For a connected almost contact metric manifold of dimension $2n+1$ with $n>2$, we have:

1. $(i)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(6)})\neq(0,0)$, then it is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$. 2. $(ii)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(6)})\neq(0,0)$, then it is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$. 3. $(iii)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(6)})\neq(0,0)$, then it is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. 4. $(iv)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(6)})\neq(0,0)$, then it is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. 5. $(v)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(7)})\neq(0,0)$, then it is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$. 6. $(vi)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(7)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$. 7. $(vii)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(7)})\neq(0,0)$, then it is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. 8. $(viii)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(7)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. 9. $(ix)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $(\xi_{(6)},\xi_{(8)})\neq(0,0)$, then it is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$. 10. $(x)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $(\xi_{(6)},\xi_{(8)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$. 11. $(xi)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(6)},\xi_{(8)})\neq(0,0)$, then it is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. 12. $(xii)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(6)},\xi_{(8)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. 13. $(xiii)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(6)},\xi_{(11)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. 14. $(xiv)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(6)},\xi_{(11)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. 15. $(xv)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(10)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. 16. $(xvi)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(10)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$. 17. $(xvii)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $(\xi_{(6)},\xi_{(10)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$. 18. $(xviii)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(12)})\neq(0,0)$ and $dd^{*}\eta$ is proportional to $\eta$, then it is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{12}$. 19. $(xix)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(12)})\neq(0,0)$ and $dd^{*}\eta$ is proportional to $\eta$, then it is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{7}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{12}$. 20. $(xx)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}$ with $(\xi_{(4)},\xi_{(6)})\neq(0,0)$ and $d(d^{*}F(\zeta))$ is proportional to $\eta$, then it is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}$. 21. $(xxi)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(6)},\xi_{(12)})\neq(0,0)$ and $d(d^{*}F(\zeta))$ is proportional to $\eta$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. 22. $(xxii)$

If the structure is of type $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(6)})\neq(0,0)$ and $\langle d\xi_{\zeta}\eta,F\rangle=0$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$.

Proof.

For $(i)$, $(ii)$, $(iii)$ and $(iv)$, making use of Proposition 4.2, we obtain $d^{*}\eta d^{*}F(\zeta)=0$. Since $(d^{*}\eta,d^{*}F(\zeta))\neq(0,0)$, it follows that set $A$ of those points such that $d^{*}F(\zeta)=0$ coincides with set of points such that $d^{*}\eta\neq 0$. Hence $A$ is open. Likewise the set $B$ consisting of points such that $d^{*}\eta=0$ coincides with the set of points such that $d^{*}F(\zeta)\neq 0$. Thus $B$ is open. Since the manifold is connected, $A$ or $B$ must be empty. Therefore, $d^{*}F(\zeta)=0$ on the whole manifold, or $d^{*}\eta=0$ on the whole manifold.

For $(v)$, $(vi)$, $(vii)$ and $(viii)$, also making use of Proposition 4.2, we obtain $d^{*}\eta(\xi_{(7)X}\eta)(Y)=0$. Since $(d^{*}\eta,\xi_{(7)})\neq(0,0)$, the proof is derived by a similar argument as before.

For $(ix)$, $(x)$, $(xi)$ and $(xii)$, again making use of Proposition 4.2, $d^{*}F(\zeta)(\xi_{(8)X}\eta)(Y)=0$ is obtained. Since $(d^{*}F(\zeta),\xi_{(8)})\neq(0,0)$, the proof is similar as before. Additionally, part $(xvii)$ is used for $(x)$ and part $(xiv)$ is used for $(xii)$.

For $(xiii)$ and $(xix)$, now we make use of Proposition 4.4 and obtain $d^{*}F(\zeta)\langle\xi_{(11)\zeta}X,Y\rangle=0$. Since $(d^{*}F(\zeta),\xi_{(11)})\neq(0,0)$, the proof is similar as before. Additionally, parts $(ii)$ and $(x)$ are used for $(xiii)$ and part $(xvii)$ is used for $(xiv)$.

For $(xv)$ and $(xvi)$, by Proposition 4.4, we obtain $d^{*}F(\zeta)\langle\xi_{(11)\zeta}X,Y\rangle=0$. Since $(d^{*}F(\zeta),\xi_{(11)})\neq(0,0)$, the proof is similar as before. Additionally, part $(vi)$ is used for $(xvi)$.

For $(xvii)$, we also make use of Proposition 4.4 and obtain $d^{*}F(\zeta)(\xi_{(10)X}\eta)(Y)=0$. Since $(d^{*}F(\zeta),\xi_{(10)})\neq(0,0)$, the proof is similar as in previous cases.

For $(xviii)$ and $(xix)$, we make use of Proposition 4.6 and obtain $d^{*}\eta(\xi_{(12)\zeta}\eta)(X)=0$. Since $(d^{*}\eta,\xi_{(12)})\neq(0,0)$, the proof is similar as in previous cases together with the use of $(i)$.

For $(xxi)$, we make use of Lemma 4.7 and obtain $d^{*}F(\zeta)\theta(X)=0$. Since $(d^{*}F(\zeta),\theta)\neq(0,0)$, the proof is similar as in previous cases together with the use of $(i)$.

For $(xxi)$, we make use of Lemma 4.7 and obtain $d^{*}F(\zeta)(\xi_{(12)\zeta}\eta)(X)=0$. The proof is similar as in previous cases using $(d^{*}F(\zeta),\xi_{(12)})\neq(0,0)$, $(i)$ and $(xxii)$.

For $(xxii)$, this case is essentially a further conclusion that the one obtained in part $(iv)$ and it was already proved in detail in [13]. ∎

Remark 6.2**.**

For connected almost contact metric manifolds of dimensi n $2n+1$, $n>2$, by Theorem 6.1, we have:

By part $(i)$, the non-existence of structures of types $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $\xi_{(5)}\neq 0$ and $\xi_{(6)}\neq 0$ implies that $2^{7}$ types do not exist. 2. -

By part $(ii)$, the non-existence of structures of types $\mathcal{C}_{1}\oplus\mathcal{C}_{2}\oplus\mathcal{C}_{3}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $\xi_{(5)}\neq 0$ and $\xi_{(6)}\neq 0$ implies that another $2^{7}-2^{6}=2^{6}$ types do not exist. 3. -

Part $(iii)$ implies that another $2^{7}-2^{6}=2^{6}$ types do not exist. 4. -

Part $(iv)$ implies that another $2^{5}$ types do not exist. 5. -

Part $(v)$ implies that another $2^{4}$ types do not exist. 6. -

Part $(vi)$ implies that another $2^{3}$ types do not exist. 7. -

Part $(vii)$ implies that another $2^{3}$ types do not exist. 8. -

Part $(viii)$ implies that another $2^{2}$ types do not exist. 9. -

Part $(ix)$ implies that another $2^{4}$ types do not exist. 10. -

Part $(x)$ implies that another $2^{3}+2^{3}$ types do not exist. 11. -

Part $(xi)$ implies that another $2^{3}$ types do not exist. 12. -

Part $(xii)$ implies that another $2^{3}+2^{3}$ types do not exist. 13. -

The non-existence of types due to part $(xiii)$ has already considered in the previous cases. 14. -

Part $(xiv)$ implies that another $2^{3}$ types do not exist. They are the ones such that $\xi_{(6)}\neq 0$, $\xi_{(10)}\neq 0$ and $\xi_{(11)}\neq 0$. 15. -

Part $(xv)$ implies that another $2^{4}$ types do not exist. 16. -

Part $(xvi)$ implies that another $2^{2}$ types do not exist. 17. -

Part $(xvii)$ implies the non-existence of types already considered in part $(x)$. 18. -

Part $(xxii)$ in case $\xi_{(12)}=0$ implies the non-existence of another $2^{2}$ types.

All together implies the non-existence of $412$ types on a connected almost contact metric manifold of dimension $2n+1$ with $n>2$. Therefore, the possible types for higher dimensions are $4096-412=3684$, where the number $4096=2^{12}$ arises at the beginning by considering the possible types from algebraic point of view. Later, due to geometry, some of these types can not exist on a connected manifold. **

Theorem 6.3**.**

For a connected almost contact metric manifold of dimension $5$, we have:

1. $(i)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(6)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$. 2. $(ii)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(6)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$. 3. $(iii)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(6)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. 4. $(iv)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(6)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$. 5. $(v)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(6)},\xi_{(11)})\neq(0,0)$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. 6. $(vi)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{12}$ with $(\xi_{(5)},\xi_{(12)})\neq(0,0)$ and $dd^{*}\eta$ is proportional to $\eta$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{12}$. 7. $(vii)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}$ with $(\xi_{(4)},\xi_{(6)})\neq(0,0)$ and $d(d^{*}F(\zeta))$ is proportional to $\eta$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{4}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}$. 8. $(viii)$

If the structure is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$ with $(\xi_{(6)},\xi_{(12)})\neq(0,0)$ and $d(d^{*}F(\zeta))$ is proportional to $\eta$, then it is of type $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}$ or $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$.

Proof.

It follows in a similar way as the proof for Theorem 6.1 in the particular case of $n=2$. Note that for $n=2$ some cases of such a theorem do not appear or are redundant. ∎

Remark 6.4**.**

On a connected almost contact metric manifold of dimensi n $5$ by using Theorem 6.3, we have:

By part $(i)$, the non-existence of structures of types $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{7}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $\xi_{(5)}\neq 0$ and $\xi_{(6)}\neq 0$ implies that $2^{5}$ types do not exist. Also, if $\xi_{(5)}\neq 0$, then the type is $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$. This implies that another $2^{4}$ types do no exist. 2. -

By part $(ii)$, the non-existence of structures of types $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{8}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$ with $\xi_{(5)}\neq 0$ and $\xi_{(6)}\neq 0$ implies that another $2^{5}-2^{4}=2^{4}$ types do not exist. Also, because if $\xi_{(6)}\neq 0$ then the type is $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{10}\oplus\mathcal{C}_{12}$, $2^{4}$ types do not exist. 3. -

Part $(iii)$ implies that another $2^{5}-2^{4}=2^{4}$ types do not exist. Also, in this case, if $\xi_{(5)}\neq 0$, then the type is $\mathcal{C}_{2}\oplus\mathcal{C}_{5}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{11}\oplus\mathcal{C}_{12}$. This implies that another $2^{3}$ types do no exist which have not been considered yet. 4. -

Part $(iv)$ implies that another $2^{4}-2^{3}=2^{3}$ types do not exist. If $\xi_{(6)}\neq 0$, then the type is $\mathcal{C}_{2}\oplus\mathcal{C}_{6}\oplus\mathcal{C}_{9}\oplus\mathcal{C}_{12}$. This implies that another $3.2^{3}=24$ types do not exist. 5. -

Part $(v)$ implies the non-existence of $2^{3}$ types which have not been considered yet.

All together implies the non-existence of $144$ types of connected almost contact metric manifold of dimensi n $5$. Therefore, the possible types in dimension $5$ are $1024-144=880$, where the number $1024=2^{8}$ arises by considering the possible types from algebraic point of view.

Note that there is not any restriction for dimension $3$. The possible types in this case are as it is expected from algebraic point of view, i.e. $2^{4}=16$. **