Geometry of Kenmotsu manifolds was originated by Kenmotsu [13] and became an interesting area of research in differential geometry. As a generalization of Kenmotsu manifolds, the notion of almost Kenmotsu manifolds was first introduced by Janssens and Vanhecke [12]. In recent years, some results regarding such manifolds we refer the reader to [5, 6, 7, 8, 9, 10, 13, 15, 16, 17, 18, 19, 21, 22]. Almost Kenmotsu manifolds satisfying the (k, μ) and (k, μ)′-nullity conditions were introduced by Dileo and Pastore [10], where k and μ both are constants. In 2011, Pastore and Saltarelli in [14] extend the above nullity conditions to the corresponding generalized nullity conditions for which both k and μ are smooth functions. Recently some results on generalized (k, μ) and (k, μ)′-almost Kenmotsu manifolds satisfying some conditions are obtained by Wang et al. [20, 21].

Gray [11] introduced two classes of Riemannian manifolds determined by the covariant derivative of the Ricci tensor; the class A consisting of all Riemannian manifolds whose Ricci tensor S is of Codazzi type, that is,

for all smooth vector fields X, Y,Z.

The class B consisting all Riemannian manifolds whose Ricci tensor S is cyclic parallel, that is,

for all smooth vector fields X, Y,Z.

A Riemannian manifold is said to be harmonic Weyl tensor if div C = 0, where C is the Weyl conformal curvature tensor of type (1, 3) defined by [22],

Q is the Ricci operator defined by S(X, Y) = g(QX, Y ), r is the scalar curvature of the manifold and ‘div’ denotes divergence. If div C = 0, then we get

A Riemannian manifold is said to be harmonic if div R = 0, which is equivalent to

for all smooth vector fields X, Y,Z.

Recently Wang et al. [15] studied conformally flat almost Kenmotsu manifolds with ξ belonging to the generalized (k, μ)′-nullity distribution. In 2016, Wang [20] studied cyclic parallel Ricci tensor in such a manifold. Moreover in [17] Wang et al. studied φ-recurrent almost Kenmotsu manifold with generalized (k, μ)′-nullity distribution.

Motivated by the above studies in the present paper we study certain curvature conditions in generalized (k, μ)′-almost Kenmotsu manifolds.

The present paper is organized as follows:

In Section 2, we first recall some basic formulas of almost Kenmotsu manifolds, while Section 3 contains some well-known results on almost Kenmotsu manifolds with generalized (k, μ)′-nullity distribution. Section 4 is devoted to study Codazzi type of Ricci tensor in such a manifold. Next in Section 5 we study almost Kenmotsu manifolds with generalized (k, μ)′-nullity distribution satisfying div C = 0. Finally, we study cyclic parallel Ricci tensor on an almost Kenmotsu manifold with generalized (k, μ)′-nullity distribution.

A differentiable (2n + 1)-dimensional manifold M is said to have a (φ, ξ, η)-structure or an almost contact structure, if it admits a (1, 1) tensor field φ, a characteristic vector field ξ and a 1-form η satisfying [1, 2],

where I denotes the identity endomorphism. Here also φξ = 0 and η ∘ φ = 0; both can be derived from (2.1) easily.

If a manifold M with a (φ, ξ, η)-structure admits a Riemannian metric g such that

for any vector fields X, Y of T_pM²ⁿ⁺¹, then M is said to have an almost contact metric structure (φ, ξ, η, g). The fundamental 2-form Φ on an almost contact metric manifold is defined by Φ(X, Y ) = g(X, φY ) for any X, Y of T_pM²ⁿ⁺¹. The condition for an almost contact metric manifold being normal is equivalent to vanishing of the (1, 2)-type torsion tensor N_φ, defined by N_φ = [φ, φ] + 2dη ⊗ ξ, where [φ, φ] is the Nijenhuis torsion of φ. Recently in [9, 10, 20], almost contact metric manifold such that η is closed and dΦ = 2η ∧ Φ are studied and they are called almost Kenmotsu manifolds. Obviously, a normal almost Kenmotsu manifold is a Kenmotsu manifold. Also Kenmotsu manifolds can be characterized by (∇_Xφ)Y = g(φX, Y )ξ−η(Y )φX, for any vector fields X, Y. It is well known [13] that a Kenmotsu manifold M²ⁿ⁺¹ is locally a warped product I ×_f N_2n, where N²ⁿ is a Kähler manifold, I is an open interval with coordinate t and the warping function f, defined by f = ce^t for some positive constant c. Let us denote the distribution orthogonal to ξ by and defined by . In an almost Kenmotsu manifold, since η is closed, is an integrable distribution. Let M²ⁿ⁺¹ be an almost Kenmotsu manifold. We denote by h=12ℒξφ and l = R(·, ξ)ξ on M²ⁿ⁺¹. The tensor fields l and h are symmetric operators and satisfy the following relations [10]:

for any vector fields X, Y. The (1, 1)-type symmetric tensor field h′ = h ∘ φ is anti-commuting with φ and h′ξ = 0. Also it is clear that [3, 10, 21]

This section is devoted to study almost Kenmotsu manifolds with ξ belonging to the generalized (k, μ)′-nullity distribution. Let M²ⁿ⁺¹(φ, ξ, η, g) be an almost Kenmotsu manifold with ξ belonging to the generalized (k, μ)′-nullity distribution, then according to Pastore and Saltarelli [14] we have

where k, μ are smooth functions on M²ⁿ⁺¹ and h′ = h ∘ φ. Let be the eigenvector of h′ corresponding to the eigenvalue λ. Then from (2.6) it is clear that λ² = −(k + 1). Therefore k ≤ −1 and λ=±-k-1. We denote by [λ]′ and [−λ]′ the corresponding eigenspaces related to the non-zero eigen value λ and −λ of h′, respectively. In [14] Pastore and Saltarelli cited some examples of almost Kenmotsu manifold with ξ belonging to the generalized (k, μ)′-nullity distribution. Before presenting our main theorems we recall some results:

Lemma 3.1.(Theorem 5.1 of [14])

Let (M²ⁿ⁺¹, φ, ξ, η, g) be a generalized (k, μ)′-almost Kenmotsu manifold such that h′ ≠ 0 and n > 1. Then for any X_λ, Y_λ, Z_λ ∈ [λ]′ and X_−λ, Y_−λ, Z_−λ ∈ [−λ]′, the Riemannian curvature tensor satisfies

If n > 1, then the Ricci operator Q of M²ⁿ⁺¹ defined by g(QX, Y ) = S(X, Y ) is given by [20]

Moreover, the scalar curvature of M²ⁿ⁺¹ is 2n(k − 2n).

Also for an almost Kenmotsu manifold with generalized (k, μ)′-nullity distribution [14]

for all smooth vectors fields X, Y.

From (3.1) it follows that

Contracting X in (3.1) we have

In this section we characterize an almost Kenmotsu manifold with ξ belonging to the generalized (k, μ)′-nullity distribution whose Ricci tensor is of Codazzi type.

Taking covariant differentiation of (3.2) we obtain

Using (2.1), (2.3) and (3.3) in (4.1) yields

Suppose the Ricci tensor of the manifold M²ⁿ⁺¹ is of Codazzi type. Then

for all smooth vector fields X, Y.

Making use of (4.2) in (4.3) implies

Using the fact h′² = (k + 1)φ² in (4.4) we have

Putting Y = ξ in the foregoing equation yields

Let X ∈ [λ]′. Then from (4.6) we get

Again assume that X ∈ [−λ]′. Then from (4.6) we obtain

Adding (4.7) and (4.8) we get

that is, either k = −1 or μ = −2. If k = −1, then h′ = 0, which is a contradiction. Therefore, μ = −2. Putting μ = −2 in (4.7) yields

Using the fact λ² = −(k + 1) in (4.9), we have

that is, either λ = 0 or λ² = 1. If λ = 0, then h′ = 0, which is a contradiction. Therefore, λ² = 1. Making use of λ² = 1 in λ² = −(k + 1) implies k = −2. Then we have from Lemma 3.1,

and

for any vector field X_λ, Y_λ, Z_λ ∈ [λ]′ and X_−λ, Y_−λ, Z_−λ ∈ [−λ]′. Also noticing μ = −2, it follows from Lemma 3.1 that K(X, ξ) = −4 for any X ∈ [λ]′ and K(X, ξ) = 0 for any X ∈ [−λ]′. Again from Lemma 3.1, we see that K(X, Y ) = −4 for any X, Y ∈ [λ]′; K(X, Y ) = 0 for any X, Y ∈ [−λ]′ and K(X, Y) = 0 for any X ∈ [λ]′, Y ∈ [−λ]′. Also the distribution [ξ] ⊕ [λ]′ is integrable with totally geodesic leaves and the distribution [−λ]′ is integrable with totally umbilical leaves by H = −(1−λ)ξ, where H is the mean curvature vector field for the leaves of [−λ]′ immersed in M²ⁿ⁺¹. Here λ = 1, then two orthogonal distributions [ξ] ⊕ [λ]′ and [−λ]′ are both integrable with totally geodesic leaves immersed in M²ⁿ⁺¹. Then we can say that M²ⁿ⁺¹ is locally isometric to Hⁿ⁺¹(−4) × ℝⁿ.

Conversely, let M²ⁿ⁺¹ is locally isometric to Hⁿ⁺¹(−4) × ℝⁿ. Then by Theorem 4.4 of [20] it follows that M²ⁿ⁺¹ is locally symmetric. Then the manifold satisfies the condition of Codazzi type of Ricci tensor, that is, (∇_Y Q)X = (∇_XQ)Y, for all smooth vector fields X, Y.

Thus we have the following:

Proposition 4.1

Let M²ⁿ⁺¹(φ, ξ, η, g) be an almost Kenmotsu manifolds with generalized (k, μ)′-nullity distribution such that h′ ≠ 0 and n > 1. Then the Ricci tensor of the manifold is of Codazzi type if and only if the manifold is locally isometric to Hⁿ⁺¹(−4) × ℝⁿ.

Let M²ⁿ⁺¹ be an almost Kenmotsu manifold whose Weyl conformal curvature tensor is divergence free, that is, div C = 0. Then we have

Using (3.2) in (5.1) yields

Using (3.5) and the fact h² = (k + 1)φ² in (5.2) implies

Replacing Y by ξ in (5.3) we have

Let X,Z ∈ D[λ]′. Then from (5.4) we obtain

Again we assume X,Z ∈ [−λ]′. Then (5.4) implies

Adding (5.5) and (5.6) we have

that is, either k = −1 or μ = −2. If k = −1, then h′ = 0, which is a contradiction. Therefore, μ = −2. Making use of μ = −2 in (5.5) yields

Using the fact λ² = −(k + 1) in (5.7), we have

that is, either λ = 0 or λ² = 1. If λ = 0, then h′ = 0, which is a contradiction. Therefore, λ² = 1. Making use of λ² = 1 in λ² = −(k + 1), implies k = −2. Then we have from Lemma 3.1,

and

for any vector field X_λ, Y_λ, Z_λ ∈ [λ]′ and X_−λ, Y_−λ, Z_−λ ∈ [−λ]′. Also noticing μ = −2, it follows from Lemma 3.1 that K(X, ξ) = −4 for any X ∈ [λ]′ and K(X, ξ) = 0 for any X ∈ [−λ]′. Again from Lemma 3.1, we see that K(X, Y ) = −4 for any X, Y ∈ [λ]′; K(X, Y ) = 0 for any X, Y ∈ [−λ]′ and K(X, Y) = 0 for any X ∈ [λ]′, Y ∈ [−λ]′. Also the distribution [ξ] ⊕ [λ]′ is integrable with totally geodesic leaves and the distribution [−λ]′ is integrable with totally umbilical leaves by H = −(1−λ)ξ, where H is the mean curvature vector field for the leaves of [−λ]′ immersed in M²ⁿ⁺¹. Here λ = 1, then two orthogonal distributions [ξ] ⊕ [λ]′ and [−λ]′ are both integrable with totally geodesic leaves immersed in M²ⁿ⁺¹. Then we can say that M²ⁿ⁺¹ is locally isometric to Hⁿ⁺¹(−4) × ℝⁿ.

This leads to the following:

Proposition 5.1

Let M²ⁿ⁺¹(φ, ξ, η, g) be an almost Kenmotsu manifold with generalized (k, μ)′-nullity distribution such that h′ ≠ 0 and n > 1. If the manifold is of harmonic Weyl conformal curvature tensor, that is, div C = 0, then the manifold is locally isometric to Hⁿ⁺¹(−4) × ℝⁿ.

Since div R = 0 implies div C = 0, thus we have the following:

Corollary 5.1

Let M²ⁿ⁺¹(φ, ξ, η, g) be an almost Kenmotsu manifold with generalized (k, μ)′-nullity distribution such that h′ ≠ 0 and n > 1. If the manifold satisfies div R = 0, then the manifold is locally isometric to Hⁿ⁺¹(−4) × ℝⁿ.

Again, since ∇C = 0 (conformally symmetric) implies div C = 0, so we obtain the following:

Corollary 5.2

Conformally symmetric almost Kenmotsu manifolds with generalized (k, μ)′-nullity distribution such that h′ ≠ 0 and n > 1 is locally isometric to Hⁿ⁺¹(−4) × ℝⁿ.

Remark 5.1

The above Corollary have been proved by De et al. [4].

Suppose the Ricci tensor of the manifold is of Codazzi type. Then the scalar curvature r is constant. Now if r is constant then from (5.1) it clear that div C = 0.

From Proposition 4.1, Proposition 5.1 and the above discussions we can state the following:

Theorem 5.1

Let M²ⁿ⁺¹(φ, ξ, η, g) be an almost Kenmotsu manifold with generalized (k, μ)′-nullity distribution such that h′ ≠ 0 and n > 1. Then the following statements are equivalent:

• (i) The Ricci tensor of M²ⁿ⁺¹is of Coddazi type,

• (ii) The manifold M²ⁿ⁺¹satisfies div C = 0,

• (iii) The manifold M²ⁿ⁺¹is locally isometric to Hⁿ⁺¹(−4) × ℝⁿ.

In this section we study almost Kenmotsu manifolds with generalized (k, μ)′-nullity distribution satisfying cyclic parallel Ricci tensor. Suppose the manifold M⁽²ⁿ⁺¹⁾ satisfies cyclic parallel Ricci tensor. Then

Taking inner product with Z in (4.2) yields

Using (6.2) in (6.1) yields

Replacing Z by ξ in (6.3) we obtain

Let X, Y ∈ [λ]′. Then from (6.4) we have

Now we assume that X, Y ∈ [−λ]′. Then from (6.4) it follows that

Using (6.5) and (6.6) implies

that is, either k = −1 or μ = −2. If k = −1, then h′ = 0, which is a contradiction. Therefore, μ = −2. Making use of μ = −2 in (6.5) yields

Using the fact λ² = −(k + 1) in (6.7), we have

that is, either λ = 0 or λ² = 1. If λ = 0, then h′ = 0, which is a contradiction. Therefore, λ² = 1. Making use of λ² = 1 in λ² = −(k + 1), implies k = −2. Then we have from Lemma 3.1,

and

for any vector field X_λ, Y_λ, Z_λ ∈ [λ]′ and X_−λ, Y_−λ, Z_−λ ∈ [−λ]′. Also noticing μ = −2, it follows from Lemma 3.1 that K(X, ξ) = −4 for any X ∈ [λ]′ and K(X, ξ) = 0 for any X ∈ [−λ]′. Again from Lemma 3.1, we see that K(X, Y ) = −4 for any X, Y ∈ [λ]′; K(X, Y ) = 0 for any X, Y ∈ [−λ]′ and K(X, Y) = 0 for any X ∈ [λ]′, Y ∈ [−λ]′. Also the distribution [ξ] ⊕ [λ]′ is integrable with totally geodesic leaves and the distribution [−λ]′ is integrable with totally umbilical leaves by H = −(1−λ)ξ, where H is the mean curvature vector field for the leaves of [−λ]′ immersed in M²ⁿ⁺¹. Here λ = 1, then two orthogonal distributions [ξ] ⊕ [λ]′ and [−λ]′ are both integrable with totally geodesic leaves immersed in M²ⁿ⁺¹. Then we can say that M²ⁿ⁺¹ is locally isometric to Hⁿ⁺¹(−4) × ℝⁿ.

Thus we can state:

Theorem 6.1

Let M²ⁿ⁺¹(φ, ξ, η, g) be an almost Kenmotsu manifolds with generalized (k, μ)′-nullity distribution such that h′ ≠ 0 and n > 1. If the manifold satisfies the cyclic parallel Ricci tensor, then the manifold is locally isometric to Hⁿ⁺¹(−4) × ℝⁿ.