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# A class of 3-dimensional almost Kenmotsu manifolds with harmonic curvature tensors

Yaning Wang
• Corresponding author
• Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, School of Mathematics and Information Sciences, Henan Normal University, Xinxiang 453007, Henan, China
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Published Online: 2016-12-10 | DOI: https://doi.org/10.1515/math-2016-0088

## Abstract

Let M3 be a three-dimensional almost Kenmotsu manifold satisfying ▽ξh = 0. In this paper, we prove that the curvature tensor of M3 is harmonic if and only if M3 is locally isometric to either the hyperbolic space ℍ3(-1) or the Riemannian product ℍ2(−4) × ℝ. This generalizes a recent result obtained by [Wang Y., Three-dimensional locally symmetric almost Kenmotsu manifolds, Ann. Polon. Math., 2016, 116, 79-86] and [Cho J.T., Local symmetry on almost Kenmotsu three-manifolds, Hokkaido Math. J., 2016, 45, 435-442].

MSC 2010: 53D15; 53C25

## 1 Introduction

Let M2n+1 be a (2n + 1)-dimensional smooth differentiable manifold on which there exists an almost contact metric structure (ϕ, ξ, η, g) defined by $ϕ2=−id+η⊗ξ,ηξ=1,gϕX,ϕY=gX,Y−ηXηY,$ where id denotes the identity map, ϕ a (1, 1)-type tensor field, ξ a global vector field tangent to M2n+1, η a global one form and g a Riemannian metric. Generally, ξ is said to be the Reeb or characteristic vector field and η is a contact 1-form. We denote by (M2n+1, ϕ, ξ, η, g) a smooth manifold furnished with an almost contact metric structure and it is said to be an almost contact metric manifold. The fundamental 2-form Φ of an almost contact metric manifold is defined by $ΦX,Y=gX,ϕY$ for any vector fields X, Y. An almost contact metric manifold turns out to be a contact metric manifold if dη = Φ (see [3]) or an almost Kenmotsu manifold if dη = 0 and dΦ = 2η ∧ Φ (see [4]). On the Riemannian product of an almost contact metric manifold M2n+1 and ℝ, there exists an almost complex structure defined by $JX,fddt=ϕX−fξ,ηXddt,$ where X denotes a vector field tangent to M2n+1, t is the coordinate of ℝ and f is a C-function on M2n+1 × ℝ. We denote by [ϕ, ϕ] the the Nijenhuis tensor of ϕ, if [ϕ, ϕ] = −2dη⊗ ξ (or equivalently, the almost complex structure J is integrable), then the almost contact metric structure is said to be normal. A normal contact metric manifold and a normal almost Kenmotsu manifold is said to be a Sasakian and a Kenmotsu manifold (see [5]) respectively. We refer the reader to Blair [3] for more details on the geometry of almost contact manifolds.

In 1972, K. Kenmotsu in [5] proved that a locally symmetric Kenmotsu manifold is of constant sectional curvature −1. Generalizing Kenmotsu’s results, Dileo and Pastore in [6] proved that a locally symmetric almost Kenmotsu manifold with R(X, Y)ξ = 0 for any vector fields X, Y orthogonal to the Reeb vector field and h ≠ 0 is locally isometric to the Riemannian product ℍn+1 (−4) × ℝn. Moreover, the present author and X. Liu in [7] proved that a locally symmetric CR-integrable almost Kenmotsu manifold of dimension greater than three is locally isometric to either the hyperbolic space ℍ2n+1 (− 1 ) or the product ℍn+1 (−4) × ℝn. Recently, the present author [1] and J. T. Cho [2] independently obtained that a three-dimensional locally symmetric almost Kenmotsu manifold is locally isometric to either the hyperbolic space ℍ3(−1) or the product ℍ2(−4) × ℝ.

In this paper, we start to study a class of three-dimensional almost Kenmotsu manifolds satisfying a reasonable geometric condition, namely ▽ξ h = 0, where 2h is the Lie differentiation of the contact structure ϕ along the Reeb vector field. We mainly obtain that the Riemannian curvature tensor of such an almost Kenmotsu manifold is harmonic if and only if the manifold is locally isometric to either the hyperbolic space ℍ3(−1) or the Riemannian product ℍ2(−4) × ℝ. We remark that this generalizes a recent result obtained by Y. Wang [1] and J. T. Cho [2] (see Corollary 4.3 for more details).

The present paper is arranged as follows. In Section 2, we collect some necessary basics and formulas on three-dimensional almost Kenmotsu manifolds. Next, in Section 3, we discuss the relations between three-dimensional Lie groups endowed with a left invariant almost Kenmotsu structure and the generalized (k, μ, v)-nullity conditions. Using these results, we present a concrete example of a three-dimensional almost Kenmotsu manifold satisfying ▽ξ h = 0 but h ≠ 0. Finally, in Section 4, we provide our main results with their proofs.

## 2 Three-dimensional Almost Kenmotsu manifolds

Let (M3, ϕ, ξ, η, g) be a 3-dimensional almost Kenmotsu manifold. In what follows, we denote by l = R(⋅, ξ) ξ, $h\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{1}{2}{\mathcal{L}}_{\xi }\varphi$ and h′ = h o ϕ, where $\mathcal{L}$ denotes the Lie differentiation and R is the Riemannian curvature tensor. From Dileo and Pastore [6, 8], we see that both h and h′ are symmetric operators and we recall some properties of almost Kenmotsu manifolds as follows: $hξ=lξ=0,trh=trh′=0,hϕ+ϕh=0,$(1) $∇ξ=h′+id−η⊗ξ,$(2) $ϕlϕ−l=2h2−ϕ2,$(3) $∇ξh=−ϕ−2h−ϕh2−ϕl,$(4) $trl=Sξ,ξ=gQξ,ξ=−2n−trh2,$(5) where S denotes the Ricci curvature tensor and Q the associated Ricci operator with respect to the metric g. Throughout this paper, we denote by 𝒟 the distribution 𝒟 = ker η which is of dimension 2n. It is easy to check that each integral manifold of 𝒟, with the restriction of ϕ, admits an almost Kähler structure. If the associated almost Kähler structure is integrable, or equivalently (see [8]), $∇XϕY=gϕX+hX,Yξ−ηYϕX+hX$(6) for any vector fields X, Y, then we say that M2n+1 is CR-integrable. Following [4, Theorem 2.1], we see that an almost Kenmotsu manifold is Kenmotsu if and only if $∇XϕY=gϕX,Yξ−ηYϕX$(7) for any vector fields X, Y. Notice that a three-dimensional almost Kenmotsu manifold is always CR-integrable. Then the following result follows immediately from (6) and (7).

Any 3-dimensional almost Kenmotsu manifold is Kenmotsu if and only if h vanishes.

Let 𝒰1 be the open subset of a 3-dimensional almost Kenmotsu manifold M3 such that h ≠ 0 and 𝒰2 the open subset of M3 which is defined by 𝒰2 ={pM3 : h = 0 in a neighborhood of p}. Therefore, 𝒰1 ∪ 𝒰2 is an open and dense subset of M3 and there exists a local orthonormal basis {ξ, e, ϕe} of three smooth unit eigenvectors of h for any point p ∊ 𝒰1 ∪ 𝒰2. On 𝒰1, we may set he = λe and hence hϕe = −λϕe, where λ is a positive function on 𝒰1. Note that the eigenvalue function λ is continuous on M3 and smooth on 𝒰1 ∪ 𝒰2.

([9, Lemma 6]). On 𝒰1 we have $∇ξξ=0,∇ξe=aϕe,∇ξϕe=−ae,∇eξ=e−λϕe=∇ee=−ξ−bϕe,∇eϕe=λξ+be,∇ϕeξ=−λe+ϕe,∇ϕee=λξ+cϕe,∇ϕeϕe=−ξ−ce,$(8) where a, b, c are smooth functions.

Applying Lemma 2.2 in the following Jacobi identity $ξ,e,ϕe+e,ϕe,ξ+ϕe,ξ,e=0$(9) yields that $eλ−ξb−ea+cλ−a−b=0,ϕeλ−ξc+ϕea+bλ+a−c=0.$(10) Moreover, applying Lemma 2.2 we have (see also [9]) the following:

On 𝒰1, the Ricci operator can be written as $Qξ=−2λ2+1ξ−ϕeλ+2λbe−eλ+2λcϕe,Qe=−ϕeλ+2λbξec+ϕeb+b2+c2+2λa+2e+ξλ+2λϕe,Qϕe=−eλ+2λcξ+ξλ+2λe−ec+ϕeb+b2+c22λa+2ϕe,$ with respect to the local basis {ξ, e, ϕe}.

## 3 Three-dimensional Lie group and some nullity conditions

Let us first recall the following definition.

A 3-dimensional almost Kenmotsu manifold is called a (k, μ, v)-almost Kenmotsu manifold if the Reeb vector field satisfies the (k, μ, v)-nullity condition, that is, $RX,Yξ=kηYX−ηXY+μηYhX−ηXhY+vηYh′X−ηXh′Y$(11) for any vector fields X, Y, where k, μ and v are smooth functions.

In the framework of almost Kenmotsu manifolds, some classes of nullity conditions were studied by many authors. We observe that a (k, μ, v)-nullity condition becomes a

• k-nullity condition if k is a constant and μ = v = 0 (see [10]);

• generalized k-nullity condition if k is a function and μ = v = 0 (see [11]);

• (k, μ)-nullity condition if k and μ are constants and v = 0 (see [8]);

• generalized (k, μ)-nullity condition if k and μ are functions and v = 0 (see [12] and [11]);

• (k, v)′-nullity condition if k and v are constants and μ = 0 (see [8]);

• generalized (k, v)′-nullity condition if k and v are functions and μ = 0 (see [12] and [11]).

Using the above definitions and some results shown in [8] we have

Any 3-dimensional non-unimodular Lie group admits a left invariant almost Kenmotsu structure for which the Reeb vector field satisfies the (k, μ, v)-nullity condition with k, μ and v being constants.

By [8, Theorem 5.2] we know that on any 3-dimensional non-unimodular Lie group there exists an almost Kenmotsu structure. Next, we recall the proof of this result shown in [8]. Let G be a 3-dimensional non-unimodular Lie group, then there exists a left invariant local orthonormal frame fields {e1, e2, e3} satisfying $e1,e2=αe2+βe3,e2,e3=0,e1,e3=γe2+δe3$(12) and a+δ = 2, where a, β, γ, δ ∊ ℝ. We define a metric g on G by g(ei, ej ) = δij for 1 ≤ i, j ≤ 3. Also, we denote by ξ = −e1 and denote by η the dual l-form of ξ. Thus, we may define a (1, 1)-type tensor field ϕ by ϕ(ξ) = 0, ϕ(e2) = e3 and ϕ(e3) = −e2. Then, one can check that (G, ϕ, ζ, η, g) admits a left invariant almost Kenmotsu structure. Next, we prove that the Reeb vector field of this almost Kenmotsu structure satisfies the (k, μ, v)-nullity condition with k, μ, v being constants. Firstly, using the Levi-Civita equation and (12) we obtain $∇ξξ=0,∇e2ξ=αe2+12β+γe3,∇e3ξ=12β+γe2+2−αe3,∇ξe2=12γ−βe3,∇e2e2=−αξ,∇e3e2=−12β+γξ,∇ξe3=12β+γe2,∇e2e3=−12β+γξ,∇e2e3=α−2ξ.$(13) Using (13), by a straightforward calculation we obtain $Re2,e3ξ=0,Re2,ξξ=−(α2+14β−γ3β−γ)e2+βα−2−αγe3,Re2,ξξ=βα−2−αγe2−(α−22+14β−γ3γ−β)e3.$(14) In view of (2), it follows from (13) that $he2=α−1e3−12β+γe2andhe3=12β+γe3+α−1e2.$(15) If ξ satisfies the (k, μ, v)-nullity condition, it follows from (11) and (15) that $Re2,e3ξ=0,Re2,ξξ=(k−12μβ−γ+vα−1)e2+(μα−1+12vβ−γ)e3,Re2,ξξ=(μα−1+12vβ−γ)e2+(k+12μβ−γ−vα−1)e3).$(16) Comparing (14) with (16) we state that there exists a unique solution for k, μ and v provided that either β + γ ≠ 0 or α ≠ 1, namely, $k=−α2+2α−14β−γ2−2,μ=β−γ,v=−2.$(17) Notice that the (k, μ, −2)-nullity condition defined by relation (17) implies that G has a non-Kenmotsu almost Kenmotsu structure if we assume that h ≠ 0 (or equivalently, either β + γ ≠ 0 or α ≠ 1).

Otherwise, if h = 0, taking into account (15) we observe that the condition β + γ = 0 and α = 1 holds. Using h = 0 in (2) gives that ▽ξ = id - η ⊗ ξ and hence R(X, Y= η(X)Yη(Y)X for any vector fields X, Y. This implies that ξ satisfies the (− 1, 0, 0)-nullity condition and by Proposition 2.1 we see that in this case G has a Kenmotsu structure. This completes the proof. ☐

The following proposition follows directly from (15) and (17).

The Reeb vector field of the non-Kenmotsu almost Kenmotsu structure defined in Theorem 3.2 satisfies the (k, v)’-nullity condition if and only if ß = γ and either α ≠ 1 or β ≠ 0.

From (13) and (15) we obtain the following:

On the non-Kenmotsu almost Kenmotsu structure defined in Theorem 3.2 there holdsξh = (ß - γ)h’.

Next, we show that under certain restrictions of k and μ the converse of the above Theorem 3.2 is true.

Any 3-dimensional non-Kenmotsu (k, μ,v)-almost Kenmotsu manifold with k a constant and μ invariant along the Reeb vector field is locally isometric to a 3-dimensional non-unimodular Lie group.

Let M3 be a 3-dimensional (k, μ, v)-almost Kenmotsu manifold with h ≠ 0 and k a constant, then 𝒰1 is a non-empty subset. It follows from (11) that the Reeb vector field ξ is an eigenvector field of the Ricci operator, i.e. Q ξ = 2. Using this in Lemma 2.3 we have λ2 = —k — 1 ≠ 0 and hence we get $b=c=0.$(18) In view of (8), by a simple computation we obtain that $Re,ξξ=−λ2+2λa+1e+2λϕe.$(19) Also, it follows from (11) that $Re,ξξ=k+λμe−λvϕe.$(20) Obviously, comparing (19) with (20) we obtain μ = —2a and v = —2. Using (18) in (10) we obtain that e(a) = ϕe(a) = 0. In view of the assumption μ invariant along ξ, we conclude that a is a constant. In this context, it follows from (8) that $ξ,e=λ+aϕe−e,e,ϕe=0,ϕe,ξ=a−λe+ϕe.$(21) A Lie group G is said to be unimodular if its left-invariant Haar measure is also right-invariant. It is wel1-known a Lie group G is unimodular if and only if the endomorphism adX $\mathfrak{g}\phantom{\rule{thinmathspace}{0ex}}\to \phantom{\rule{thinmathspace}{0ex}}\mathfrak{g}$ given by ad X (Y) = [X, Y] has trace equal to zero for any Xg, where g denotes the Lie algebra associated to G. Following Milnor [13], we state that M3 is locally isometric to a 3-dimensional non-unimodular Lie group. In fact, from (21) we see that its unimodular kernel {X ∊ g : trace(adX) = 0} is commutative and of 2-dimension and trace(adξ) = —2. This completes the proof. ☐

If the Reeb vector field ξ satisfies the (k, μ, v)-nullity condition, putting Y = ξ into (11) gives that $l=−kϕ2+μh+vh′.$(22) Using (22) in (3) and (4) we obtain h2 = (k + 1)ϕ2 and hence the following proposition is true.

On any 3-dimensional (k, μ, v)-almostKenmotsu manifold there holds that ▽ξh = μh’ — (ν + 2)h.

Note that the above proposition is in fact a generalization of Proposition 3.4.

## 4 Almost Kenmotsu manifolds with harmonic curvature tensors

A Riemannian manifold M is said to have harmonic curvature tensor if divR= 0, where R denotes the Riemannian curvature tensor. As is well known, the curvature tensor R is harmonic if and only if the associated Ricci tensor Q is of Codazzi-type, namely, $∇XQY=∇YQX$(23) for any vector fields X, Y on M.

Almost Kenmotsu manifolds with the Reeb vector field belonging to (k, μ)′-nullity distribution and harmonic curvature tensor were studied by the present author and X. Liu [ 14]. In this section, we aim to obtain a classification of three-dimensional almost Kenmotsu manifolds satisfying ▽ξh = 0 whose curvature tensors are harmonic.

By Proposition 3.6, we see that any 3-dimensional (k, 0, −2)-almost Kenmotsu manifold with k a function satisfies ▽ξh = 0. As a special case of this result, from Propositions 3.3 and 3.4 or [8, Theorem 4.1] we see that if a 3-dimensional non-unimodular Lie group G with a left invariant local orthonormal frame fields satisfies $[e1,e2]=αe2+βe3,[e2,e3]=0,[e1,e3]=βe2+(2−α)e3$(24) and either a ≠ 1 or β ≠ 0, then ▽ξh = 0 holds on the almost Kenmotsu structure defined by (24). In fact, the Reeb vector field ξ of the almost Kenmotsu structure satisfies the (k, 0, -2)-nullity condition with k a constant.

Notice that the condition ▽ξh = 0 was also used by Dileo and Pastore [15] in the study of almost Kenmotsu manifolds with η-parallel tensor field h′.

Using the well-known formula div $Q\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{1}{2}$ grad(r), we obtain from relation (23) that the following proposition is true.

The scalar curvature of a Riemannian manifold with harmonic curvature tensor is a constant.

Applying Proposition 4.1 we obtain the following main result.

A 3-dimensional almost Kenmotsu manifold satisfyingξh = 0 and having a harmonic curvature tensor is locally isometric to either the hyperbolic space3 (-1) or the product2(−4) × ℝ.

Let M3 be three-dimensional almost Kenmotsu manifold. We shall first consider the case h = 0, then by Proposition 2.1 we see that M3 is Kenmotsu and ▽ξh = 0 holds trivially. If the curvature tensor of M3 is harmonic, following Proposition 4.1 we observe that the scalar curvature of M3 is a constant. Notice that Inoguchi in [16, Proposition 3.1] proved that a three-dimensional Kenmotsu manifold of constant scalar curvature is of constant sectional curvature –1. This yields that a three-dimensional Kenmotsu manifolds with harmonic curvature tensor is of constant sectional curvature – 1.

Now let us consider a three-dimensional almost Kenmotsu manifold M3 satisfying h ≠ 0, then 𝒰1 is a nonempty subset. By applying Lemma 2.2 and a direct calculation we obtain $(∇ξh)e=ξ(λ)e+2aλϕeand(∇ξh)ϕe=−ξ(λ)ϕe+2aλe.$(25) By the assumption condition ▽ξh = 0 and (25) we have $ξ(λ)=a=0;$(26) where we have used that λ is positive on 𝒰1. In what follows, we denote f by $f=e(c)+ϕe(b)+b2+c2+2.$(27) Then, using (26) and (27) we obtain from Lemmas 2.2 and 2.3 that $(∇ξQ)ξ=−ξ(ϕe(λ)+2λb)e−ξ(e(λ)+2λc)ϕe.$(28) $(∇ξQ)e=−ξ(ϕe(λ)+2λb)ξ−ξ(f)e.$(29) $(∇ξQ)ϕe=−ξ(e(λ)+2λc)ξ−ξ(f)ϕe.$(30) $(∇eQ)ξ=2(ϕe(λ)−3λe(λ)+2λb−2λ2c)ξ+(f−2−e(ϕe(λ)+2λb)−b(e(λ)+2λc))e+(2λ3+b(ϕe(λ)+2λb)−e(e(λ)+2λc)−λf)ϕe.$(31) $(∇eQ)e=(f−2−e(ϕe(λ)+2λb)−b(e(λ)+2λc))ξ−(e(f)+2ϕe(λ))e+(e(λ)+λϕe(λ)+2λ2b−2λc)ϕe.$(32) $∇eQϕe=2λ3−fλ+bϕeλ+2λb−eeλ+2λcξ+eλ+λϕeλ−2λc+2λ2be+2λeλ+2λc−ef−4λbϕe.$(33) $∇ϕeQξ=2eλ−3λϕeλ+2λc−2λ2bξ+2λ3+ceλ+2λc−ϕeϕeλ+2λb−λfe+f−2−ϕeϕeλ+2λc−cϕeλ+2λbϕe.$(34) $∇ϕeQe=2λ3−fλ+ceλ+2λc−ϕeϕeλ+2λbξ−ϕef+4λc−2λϕeλ+2λbe+ϕeλ+λeλ+2λ2c−2λbϕe.$(35) $∇ϕeQϕe=f−2ϕeeλ+2λc−cϕeλ+2λbξ+ϕeλ+λeλ+2λ2c−2λbe−ϕef+2eλϕe.$(36) If we replace X and Y in (23) by e and ξ, respectively, then we have from (29) and (31) that $2(ϕe(λ)−3λe(λ)+2λb−2λ2c)+ξ(ϕe(λ)+2λb)=0,f−2−e(ϕe(λ)+2λb)−b(e(λ)+2λc)+ξ(f)=0,2λ3+b(ϕe(λ)+2λb)−e(e(λ)+2λc)−λf=0.$(37) Similarly, if we replace X and Y in (23) by ϕe and ξ, respectively, we obtain from (30) and (34) that$2(e(λ)−3λϕe(λ)+2λc−2λ2b)+ξ(e(λ)+2λc)=0,f−2−ϕe(e(λ)+2λc)−c(ϕe(λ)+2λb)+ξ(f)=0,2λ3+c(e(λ)+2λc)−ϕe(ϕe(λ)+2λb)−λf=0.$(38) Similarly, if we replace X and Y in (23) by e and ϕβ, respectively, we obtain from (33) and (35) that $c(e(λ)+2λc)−ϕe(ϕe(λ)+2λb)+e(e(λ)+2λc)=b(ϕe(λ)+2λb),e(λ)−λϕe(λ)+2λc−2λ2b+ϕe(f)=0,λe(λ)−ϕe(λ)+2λ2c−2λb−e(f)=0.$(39) Moreover, it follows from Lemma 2.3 that $r=−2(λ2+1)−2f.$(40) Applying again the well known formula div $Q\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\frac{1}{2}$ grad(r) we have from equations (28), (32) and (36) that $−ξ(ϕe(λ)+2λb)+3λe(λ)−ϕe(λ)+2λ2c−2λb=0,−ξ(e(λ)+2λc)+3λϕe(λ)−e(λ)+2λ2b−2λc=0,$(41) where we have used the scalar curvature r = constant, equation (40), the second terms of relations (37) and (38).

Next, using the first equation of (41) and the second equation of (41) in the first terms of (37) and (38), respectively, we obtain $3λe(λ)−ϕe(λ)+2λ2c−2λb=0,3λϕe(λ)−e(λ)+2λ2b−2λc=0,$(42) and $ξ(ϕe(λ))+2λξ(b)=0,ξ(e(λ))+2λξ(c)=0.$(43) Taking the covariant differentiation of relation (42) and using (43) and (26) we have $ξ(b)=ξ(c)=0,$(44) where we have used that λ is a positive function. Using (44) and (26) in (10) gives that $e(λ)=b−cλ,ϕe(λ)=c−bλ.$(45) Putting (45) into (42) yields that $2bλ−cλ2−c=0,2cλ−bλ2−b=0.$(46) It follows from (46) that (λ2 + 1)(b2c2) = 0 and hence we get either bc = 0 or b + c = 0. We continue the discussion with the following two cases.

Case i. Using b = c in (46) we have either b = c = 0 or λ = 1. Now we assume that b = c = 0 holds and using this in (10) we see that λ is a positive constant. By applying a = b = c = 0 in (8) we obtain the following $ξ,e=λϕe−e,e,ϕe=0,ϕe,ξ=−λe+ϕe.$

According to J. Milnor [13], we now conclude that M3 is locally isometric to a three-dimensional non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. Moreover, using a = b = 0 in equations (28)–(36) we get

$∇Q=0.$

Notice that the above relation holds on a three-dimensional Riemannian manifold if and only if the curvature tensor is parallel, i.e. the manifold is locally symmetric. We also observe that Wang [1, Theorem 3.4] and [2, Theorem 5] proved that a locally symmetric three-dimensional non-Kenmotsu almost Kenmotsu manifold is locally isometric to the Riemannian product ℍ2(−4) × ℝ.

Otherwise, if λ = 1 we obtain from (46) again b = c and using this in the last two equations of (37) we obtain $e(b)=0.$

Moreover, using λ = 1 and b = c in the last two equations of (38) we obtain

$ϕe(b)=0.$

In view of (44) we observe that both b and c are constants. Then it follows from the second equation of (37) that f = 2 + 2b2. Finally, using this in (28)–(36) gives that ▽Q = 0 and this is equivalent to the local symmetry. Then the proof follows from Wang [1, Theorem 3.4] or [2, Theorem 5].

Case ii. Now we consider the other case: b + c = 0. Using this in (46) gives that b = c = 0, where we have used that λ is positive. Moreover, putting b = c = 0 in (45) and applying (26) we see that λ is a constant. Therefore, the proof follows from Case i. ☐

On a three-dimensional locally symmetric almost Kenmotsu manifold, applying the local symmetry condition we obtain that ▽ξl = 0. Substituting X with ξ in (6) implies that ▽ξϕ = 0. Then, by taking the covariant differentiation of (3) along ξ we obtain ▽ξh2 = ▽ξh o h + h o ▽ξh = 0. It follows directly that (▽ξξh) o h + 2(▽ξh)2 + h o (▽ξξh) = 0. Also, using ▽ξl = ▽ξh = 0 and taking the covariant differentiation of (4) along ξ we obtain ▽ξξh = -2▽ξh. Therefore, it is easily seen that (▽ξh)2 = 0 and hence we get ▽ξh = 0. Then the following corollary follows directly from Theorem 4.2 and can also be regarded as a generalization of Wang [1, Theorem 3.4] and [2, Theorem 5].

Let M3 be a three-dimensional almost Kenmotsu manifold, then the following three statements are equivalent:

1. M3 is locally symmetric.

2. M3 satisfiesξh = 0 and the curvature tensor is harmonic.

3. M3 is locally isometric to either the hyperbolic space3(–1) or the Riemannian product2(−4) × ℝ.

## Acknowledgement

This work was supported by the National Natural Science Foundation of China (No. 11526080), Key Scientific Research Program in Universities of Henan Province (No. 16A110004), the Research Foundation for the Doctoral Program of Henan Normal University (No. qdl4145) and the Youth Science Foundation of Henan Normal University (No. 2014QK01).

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Accepted: 2016-10-27

Published Online: 2016-12-10

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455,

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