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

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].

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 / D 0 for any vector fields X; Y orthogonal to the Reeb vector field and h ¤ 0 is locally isometric to the Riemannian product H nC1 . 4/ R n . 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 H 2nC1 . 1/ or the product H nC1 . 4/ R n . 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 H 3 . 1/ or the product H 2 . 4/ R.
In this paper, we start to study a class of three-dimensional almost Kenmotsu manifolds satisfying a reasonable geometric condition, namely r h D 0, where 2h is the Lie differentiation of the contact structure 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 H 3 . 1/ or the Riemannian product H 2 . 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 threedimensional 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; ; /-nullity conditions. Using these results, we present a concrete example of a three-dimensional almost Kenmotsu manifold satisfying r h D 0 but h ¤ 0. Finally, in Section 4, we provide our main results with their proofs.

Three-dimensional Almost Kenmotsu manifolds
Let .M 3 ; ; ; Á; g/ be a 3-dimensional almost Kenmotsu manifold. In what follows, we denote by l D R. ; / , h D 1 2 L and h 0 D h ı , where 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 0 are symmetric operators and we recall some properties of almost Kenmotsu manifolds as follows: (1) tr.l/ D S. ; / D g.Q ; / D 2n trh 2 ; 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 D ker Á which is of dimension 2n. It is easy to check that each integral manifold of D, with the restriction of , admits an almost Kähler structure. If the associated almost Kähler structure is integrable, or equivalently (see [8]), 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).  e. / .c/ C e.a/ C b. C a/ c D 0: Moreover, applying Lemma 2.2 we have (see also [9]) the following:

Three-dimensional Lie group and some nullity conditions
Let us first recall the following definition.
Definition 3.1. A 3-dimensional almost Kenmotsu manifold is called a .k; ; /-almost Kenmotsu manifold if the Reeb vector field satisfies the .k; ; /-nullity condition, that is, for any vector fields X; Y , where k, and 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; ; /-nullity condition becomes a k-nullity condition if k is a constant and D D 0 (see [10]); -generalized k-nullity condition if k is a function and D D 0 (see [11]); -.k; /-nullity condition if k and are constants and D 0 (see [8]); -generalized .k; /-nullity condition if k and are functions and D 0 (see [12] and [11]); -.k; / 0 -nullity condition if k and are constants and D 0 (see [8]); -generalized .k; / 0 -nullity condition if k and are functions and D 0 (see [12] and [11]).
Using the above definitions and some results shown in [8] we have Theorem 3.2. Any 3-dimensional non-unimodular Lie group admits a left invariant almost Kenmotsu structure for which the Reeb vector field satisfies the .k; ; /-nullity condition with k, and being constants.
Proof. 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 fe 1 ; e 2 ; e 3 g satisfying and˛Cı D 2, where˛;ˇ; ; ı 2 R. We define a metric g on G by g.e i ; e j / D ı ij for 1 Ä i; j Ä 3. Also, we denote by D e 1 and denote by Á the dual 1-form of . Thus, we may define a .1; 1/-type tensor field by . / D 0, .e 2 / D e 3 and .e 3 / D e 2 . Then, one can check that .G; ; ; Á; g/ admits a left invariant almost Kenmotsu structure. Next, we prove that the Reeb vector field of this almost Kenmotsu structure satisfies the .k; ; /-nullity condition with k, , being constants. Firstly, using the Levi-Civita equation and (12) we obtain Using (13), by a straightforward calculation we obtain R.e 2 ; e 3 / D 0; In view of (2), it follows from (13) that If satisfies the .k; ; /-nullity condition, it follows from (11) and (15) that R.e 2 ; e 3 / D 0; Comparing (14) with (16) we state that there exists a unique solution for k, and provided that eitherˇC ¤ 0 or˛¤ 1, namely, Notice that the .k; ; 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, eitherˇC ¤ 0 or˛¤ 1). Otherwise, if h D 0, taking into account (15) we observe that the conditionˇC D 0 and˛D 1 holds. Using h D 0 in (2) gives that r D id Á˝ and hence R.X; Y / D Á.X /Y Á.Y /X for any vector fields X; Y . This implies that 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). From (13) and (15) we obtain the following: Next, we show that under certain restrictions of k and the converse of the above Theorem 3.2 is true.
In view of (8), by a simple computation we obtain that R.e; / D . 2 C 2 a C 1/e C 2 e: Also, it follows from (11) that R.e; / D .k C /e e: Obviously, comparing (19) with (20) we obtain D 2a and D 2. Using (18) in (10) we obtain that e.a/ D e.a/ D 0. In view of the assumption invariant along , we conclude that a is a constant. In this context, it follows from (8) A Lie group G is said to be unimodular if its left-invariant Haar measure is also right-invariant. It is well-known a Lie group G is unimodular if and only if the endomorphism ad X W g ! g given by ad X .Y / D OEX; Y has trace equal to zero for any X 2 g, where g denotes the Lie algebra associated to G. Following Milnor [13], we state that M 3 is locally isometric to a 3-dimensional non-unimodular Lie group. In fact, from (21) we see that its unimodular kernel fX 2 g W trace.ad X / D 0g is commutative and of 2-dimension and trace.ad / D 2. This completes the proof.
If the Reeb vector field satisfies the .k; ; /-nullity condition, putting Y D into (11) gives that Using (22) in (3) and (4) we obtain h 2 D .k C 1/ 2 and hence the following proposition is true.

Almost Kenmotsu manifolds with harmonic curvature tensors
A Riemannian manifold M is said to have harmonic curvature tensor if divR D 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, for any vector fields X; Y on M .
Almost Kenmotsu manifolds with the Reeb vector field belonging to .k; / 0 -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 r h D 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 r h D 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 OEe 1 ; e 2 D˛e 2 Cˇe 3 ; OEe 2 ; e 3 D 0; OEe 1 ; e 3 Dˇe 2 C .2 ˛/e 3 (24) and either˛¤ 1 orˇ¤ 0, then r h D 0 holds on the almost Kenmotsu structure defined by (24). In fact, the Reeb vector field of the almost Kenmotsu structure satisfies the .k; 0; 2/-nullity condition with k a constant. Notice that the condition r h D 0 was also used by Dileo and Pastore [15] in the study of almost Kenmotsu manifolds with Á-parallel tensor field h 0 .
Using the well-known formula divQ D 1 2 grad.r/, we obtain from relation (23) that the following proposition is true.
If we replace X and Y in (23) by e and , respectively, then we have from (29) and (31) that .e. / C 2 c/ C 3 e. / e. / C 2 2 b 2 c D 0; where we have used the scalar curvature r D 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 .e. // C 2 .c/ D 0: Taking the covariant differentiation of relation (42) and using (43) and (26) we have where we have used that is a positive function. Using (44) and (26) in (10) gives that It follows from (46) that . 2 C 1/.b 2 c 2 / D 0 and hence we get either b c D 0 or b C c D 0. We continue the discussion with the following two cases. According to J. Milnor [13], we now conclude that M 3 is locally isometric to a three-dimensional non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. Moreover, using a D b D 0 in equations (28)-(36) we get rQ D 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 H 2 . 4/ R. Otherwise, if D 1 we obtain from (46) again b D c and using this in the last two equations of (37) we obtain e.b/ D 0: Moreover, using D 1 and b D c in the last two equations of (38) we obtain e.b/ D 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 D 2 C 2b 2 . Finally, using this in (28)-(36) gives that rQ D 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 c D 0. Using this in (46) gives that b D c D 0, where we have used that is positive. Moreover, putting b D c D 0 in (45) and applying (26) we see that 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 r l D 0. Substituting X with in (6) implies that r D 0. Then, by taking the covariant differentiation of (3) along we obtain r h 2 D r h ı h C h ı r h D 0. It follows directly that .r r h/ ı h C 2.r h/ 2 C h ı .r r h/ D 0. Also, using r l D r h D 0 and taking the covariant differentiation of (4) along we obtain r r h D 2r h. Therefore, it is easily seen that .r h/ 2 D 0 and hence we get r h D 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].
Corollary 4.3. Let M 3 be a three-dimensional almost Kenmotsu manifold, then the following three statements are equivalent: (1) M 3 is locally symmetric.
(2) M 3 satisfies r h D 0 and the curvature tensor is harmonic.
(3) M 3 is locally isometric to either the hyperbolic space H 3 . 1/ or the Riemannian product H 2 . 4/ R.