In 1923, Eisenhart [13] proved that if a positive definite Riemannian manifold (M, g) admits a second order parallel symmetric covariant tensor other than the constant multiple of metric tensor, then it is reducible. In 1926, Levy [18] proved that a second order parallel symmetric non-singular tensor in a space of constant curvature is a constant multiple of the metric tensor. Using Ricci identities, Sharma in [24] gives a global approach to the Eisenhart problem, and generalized the Levy's theorem. This problem was studied by the same author in contact geometry [25, 26, 27] on different manifolds, for example for K-contact manifolds in [26]. Since then many geometers have investigated the Eisenhart problem on various contact manifolds: nearly Sasakian[30], P-Sasakian [7, 29], f-Kenmotsu manifold [4], N(κ)-quasi Einstein manifold [6], 3-dimensional normal paracontact geometry [1], contact manifolds having non-vanishing ξ-sectional curvature [14], (κ,μ)-contact metric manifold [19], almost Kenmotsu manifolds [34] and 3-dimensional non-cosymplectic normal almost contact pseudo-metric manifold of non-vanishing ξ-sectional curvature [32].

In contact geometry, Kenmotsu manifolds, introduced by Kenmotsu in [17], are one of the important classes of manifolds. Such manifolds were observed to be normal. Let (M,φ,ξ,η,g) be an almost Kenmotsu structure (see Section 2) on a (2n+1)-dimensional differentiable manifold. The purpose of this paper is to study second order parallel tensors on M under certain conditions. Throughout the paper, we suppose that the almost Kenmotsu manifold is of dimension 2n+1. Denoting the Ricci tensor by S, the tensor 12£ξφ by h, where £ denotes the Lie differentiation, and the operator R(⋅,ξ)ξ by l, we prove the following.

Theorem 1.1.

Let α be a second order symmetric parallel tensor, and A be the (1,1)-tensor metrically equivalent to α on an almost Kenmotsu manifold M. The following hold

• (i) Aξ=α(ξ,ξ)ξ, if M has non-vanishing ξ-sectional curvature;
  

• (ii) trA=α(ξ,ξ)−tr(A(h2−2φh−φ(∇ξh)))+α(ξ,ξ)S(ξ,ξ);
  

• (iii) tr(Al)=α(ξ,ξ)S(ξ,ξ),
  

where tr denotes the trace.

Since the Ricci tensor S is a second order tensor, we have:

Corollary 1.1.

If an almost Kenmotsu manifold M is Ricci symmetric, and if M has non-vanishing ξ-sectional curvature, then

• (i) Qξ=−(2n+trh2)ξ;
  

• (ii) the scalar curvature r=‖Qξ‖2−2n−tr(h2)−tr(Q(h2−2φh−φ(∇ξh)));
  

• (iii) tr(Ql)=‖Qξ‖2,
  

where Q is the Ricci operator determined by S(X,Y)=g(QX,Y).

Theorem 1.2.

Let M be an almost Kenmotsu manifold having non-vanishing ξ-sectional curvature such that trl>−2n−2. The second order parallel tensor on M is a constant multiple of the associated metric tensor.

Corollary 1.2.

If M is an almost Kenmotsu manifold having non-vanishing ξ-sectional curvature such that trl>−2n−2 is Ricci symmetric, then it is Einsteinian.

Since trl=S(ξ,ξ)=−2n and the ξ-sectional curvature K(ξ,X)=−1 for Kenmotsu manifold, we have the following.

Corollary 1.3.

Any Ricci symmetric Kenmotsu manifold is Einsteinian.

Corollary 1.4.

An affine Killing vector field on M which has non-vanishing ξ-sectional curvature such that trl>−2n−2 is homothetic.

Blair, Koufogiorgos and Papantoniou [3] introduced (κ,μ)-nullity distributions on a contact metric manifold generalizing the notion of κ-nullity distributions by defining

for any p ∈ M, where κ,μ∈ℝ.

Corollary 1.5.

A second order parallel tensor on an almost Kenmotsu manifold with ξ∈N(κ,μ) is a constant multiple of the associated metric tensor.

The above corollary has been proved by Wang and Liu in [34]. Recently, Dileo-Pastore [12] introduced (κ,μ)′-nullity distribution on an almost Kenmotsu manifold which is defined by

for any p∈ M, where h′=h∘φ and κ,μ∈ℝ. Here we recall the following results due to Dileo-Pastore [11, 12].

Lemma 1.1.

([12, Proposition 4.1]) Let M be an almost Kenmotsu manifold such that h′≠0 with ξ∈N(κ,μ)′. Then κ<−1, μ=−2 and spect(h′)={0,λ,−λ} with 0 as simple eigenvalue and λ=−1−k.

Lemma 1.2.

([11, Theorem 6]) Let M be a locally symmetric almost Kenmotsu manifoldsuch that h′≠0 and R(X,Y)ξ=0 for any X,Y∈D, where D=ker(η). Then, M is locally isometric to ℍn+1(−4)×ℝn.

Lemma 1.3.

([12, Corollary 4.2]) Let M be an almost Kenmotsu manifold such that h′≠0 with ξ∈N(κ,μ)′. Then M is locally symmetric if and only if spect(h′)={0,1,−1} that is if and only if k=-2.

Lemma 1.4.

([12, Proposition 4.3]) Let M be an almost Kenmotsu manifold such that h′≠0 with ξ∈N(κ,μ)′. Then the ξ-sectional curvature satisfies

where [λ]′ denotes the eigenspace of h' related to the eigenvalue λ.

We use these to prove:

Theorem 1.3.

Let M be an almost Kenmotsu manifold with ξ∈N(κ,μ)′ and h′≠0. If M admits a second order parallel tensor, then either the second order parallel tensor is a constant multiple of the associated metric tensor or M is locally isometric to ℍn+1(−4)×ℝn.

In [34], Wang and Liu proved the above theorem in another way. Now waving the hypothesis non-vanishing ξ-sectional curvature in Theorem 1.2 by ∇ξh=0, we prove:

Theorem 1.4.

Let M be an almost Kenmotsu manifold with ∇ξh=0 such that trl>−2n−2. Then the second order parallel tensor on M is a constant multiple of the associated metric tensor.

In [23] Naik et al. proved that every vector field which leaves the curvature tensor invariant are Killing in a (κ,μ)′-almost Kenmotsu manifold with h′≠0 and κ≠−2. Here, as an application of Theorem 1.4, we prove the following.

Theorem 1.5.

Let M be an almost Kenmotsu manifold with ∇ξh=0 such that trl>−2n−2. Then every vector field keeping curvature tensor invariant are homothetic.

The m-Bakry-Emery Ricci tensor is a natural extension of the Ricci tensor to smooth metric measure spaces and is given by

where f is a smooth function on M and m is an integer such that 0<m≤∞. If Sfm is a constant multiple of the metric g, then the Riemannian manifold (M,g) is called m-quasi-Einstein manifold (see [5, 15] and the references therein). Now applying Theorem 1.2, Theorem 1.3 and Theorem 1.4, we deduce the following statement.

Theorem 1.6.

Let M be an almost Kenmotsu manifold either having non-vanishing ξ-sectional curvature such that trl>−2n−2 or ξ∈N(κ,μ)′ and h′≠0 or ∇ξh=0 such that trl>−2n−2. Then the m-Bakry-Emery Ricci tensor Sfm=S+Hessf−1mdf⊗df is parallel if and only if M is m-quasi-Einstein manifold.

A Ricci soliton on a Riemannian manifold (M,g) is defined by

where V is a smooth vector field and ρ is a constant. In the context of contact geometry and paracontact geometry, Ricci solitons are studied in [8, 9, 10, 20, 21, 22, 28, 31, 33]. In the similar vein as Theorem 1.6. we state the following.

Theorem 1.7.

Let M be an almost Kenmotsu manifold either having non-vanishing ξ-sectional curvature such that trl>−2n−2 or ξ∈N(κ,μ)′ and h′≠0 or ∇ξh=0 such that trl>−2n−2. Then the the second order tensor £Vg+2S is parallel if and only if M admits a Ricci soliton.

An almost contact metric structure on a (2n+1)-Riemannian manifold M is a quadruple (φ,ξ,η,g), where ⊎ is an endomorphism, ξ a global vector field, η a 1-form and g a Riemannian metric, such that

We easily obtain from (2.1) that φξ=0 and η∘φ=0 (see [2]).

A manifold M with (φ,ξ,η,g) structure is said to be an almost contact metric manifold. We define the fundamental 2-form on (M,φ,ξ,η,g) by Φ(X,Y)=g(X,φY).

If the 1-form η is closed and dΦ=2η∧Φ, then M is said to be an almost Kenmotsu manifold [16]. It is well known that a normal almost Kenmotsu manifold is a Kenmotsu manifold.

The two tensor fields l:=R(⋅,ξ)ξ and h:12£ξφ are known to be symmetric and satisfy

Further, one has the following formulas:

Now we prove the results stated in Section 1.

Proof of Theorem 1.1.

Suppose that α is a symmetric (0, 2)-tensor and A be the (1, 1)-tensor metrically equivalent to α, that is, g(AX,Y)=α(X,Y). Note that ∇α=0 implies ∇A=0, and so

Therefore

Thus we get

Setting X=Z=ξ in the above equation, we get

For W=ξ and Y=Aξ, equation (3.1) becomes

Since the ξ-sectional curvature K(ξ,X) is non-vanishing, equation (3.2) implies that Aξ=fξ, for some scalar function f on M. As g(ξ,ξ)=1, we have

This proves (i). Applying φ to (2.6) shows that

Using (3.3) in 3.1, we get

If {ei} is a local orthonormal basis, then putting W=Y=ei in the above equation and summing it over i, leads to

This gives (ii). Now plugging X by AX in 3.3 and then contracting with respect to X gives

Using this in (ii) yields (iii). This finishes the proof.

Proof of Corollary 1.1. As the Ricci tensor S is parallel, it follows from Theorem 1.1 that

Now (i) follows from (2.8). Note that from (3.4), we have

Hence (ii) and (iii) follows directly from (ii) and (iii) of Theorem 1.1.

Proof of Theorem 1.2. Let {ei,φei,ξ}i=1n be a local orthonormal basis such that hei=λiei. Then hφei=−λiφei. Hence as hξ=0, we have

But by hypothesis, trl=−2n−trh2>−2n−2, and so trh2<2. Therefore ∑ i=1nλi2<1 which means λi2<1 for each i. This fact will be used in the rest of analysis for this theorem.

Let α be a (0,2)-tensor such that ∇α=0, and A be the dual (1, 1)-type tensor which is metrically equivalent to α, that is, α(X,Y)=g(AX,Y). We will analyse the symmetric and anti-symmetric cases of second order tensor separately.

First, suppose that α is symmetric. Then from item (i) of Theorem 1.1, we get

To show α(ξ,ξ) is constant, we differentiate it along X to obtain

Now differentiating (3.5) along X yields

Replacing X by φX, it follows that

Now putting X=ei and X=φei in (3.6) respectively gives

and

Multiplying λi to (3.7) and then subtracting it with (3.8) shows

Similarly, we can find

Since λi2<1, we have A(ei)=α(ξ,ξ)ei and A(φei)=α(ξ,ξ)φei for every i=1,…,n. But Aξ=α(ξ,ξ)ξ, and so

for any X ∈ TM, that is, α is a constant multiple of g. Now suppose that α is skew-symmetric. Note that ∇α=0 implies

The skew-symmetry of α then gives

For X=A2ξ, Y=Z=ξ and W=Aξ, we find

From (3.10), the hypothesis K(ξ,X) is non-vanishing imply that A2ξ=fξ. Clearly,

One can easily verify the constancy of f by differentiating (3.11) along any vector field and getting the derivative 0. Now differentiating A2ξ=−‖Aξ‖2ξ along φX yields

Next, we take X=ei and φei successively in the above equation and argue as before in order to obtain A2ei=−‖Aξ‖2ei and A2(φei)=−‖Aξ‖2φei, for every i=1,2,…,n. So, A2X=−‖Aξ‖2X for any X orthogonal to ξ. Hence, we obtain

If ‖Aξ‖≠0, then J=‖Aξ‖−1A defines a Kaehlerian structure on M leading to a contradiction that M is odd dimensional. Thus ‖Aξ‖=0, and so Aξ=0. Differentiating it along φX gives

Now, as before we put X=ei and φei successively in the above equation to conclude that AX=0, for any X⊥ξ. Since Aξ=0, we obtain AX=0 for any X∈ TM. This completes the proof.

Proof of Corollary 1.4. A vector field V such that £V∇=0 is called an affine Killing vector field. Note that £V∇=0 is equivalent to ∇(£Vg)=0. Now the result follows from Theorem 1.2.

Proof of Corollary 1.5. Let M be an almost Kenmotsu manifold with ξ∈N(κ,μ)′, that is

for all U,V∈TM. It then follows from Dileo-Pastore [12] that k=-1 and h=0. Hence (3.12) gives trl=−2n, K(ξ,X)=−1 and so the conclusion follows from Theorem 1.2.

Proof of Theorem 1.3. Observe that, if X is an eigenvector of h with eigenvalue λ, and thus hφX=−λφX,then X+φX is an eigenvector of h' with eigenvalue −λ, while X−φX is eigenvector with eigenvalue λ. Thus, it follows that h and h' admit the same eigenvalues.

First suppose that κ≠−2. Note that, if X is such that X⊥ξ and hX=λX, then from Lemma we see λ is different from +1 and -1. Thus Lemma implies that K(ξ,X) is non-vanishing. If the second order parallel tensor α is symmetric, then part (i) of Theorem~1.1 shows that

That α(ξ,ξ) is constant, can be verified by differentiating it and getting the derivative equal to 0. Now differentiating (3.13) along X and φX, where X⊥ξ and hX=λX, gives

and

Since λ is different from +1 and -1, we obtain

for any X ∈ TM, that is, α is a constant multiple of g. Now suppose that α is skew-symmetric. Then as in the proof of Theorem 1.2, we obtain A2ξ=−‖Aξ‖2, where ‖Aξ‖2 is constant. Differentiating it along X and φX successively, where X⊥ξ and hX=λX, gives

and

Thus, we obtain

Proceeding the similar manner as in the Theorem 1.2 one gets AX=0 for any X∈ TM.

Now suppose that κ=−2 and ξ∈N(κ,μ)′, that is,

Then Lemma shows M is locally symmetric. Now it follows from Lemma that M is locally isometric to ℍn+1(−4)×ℝn, since R(X,Y)ξ=0 for any X,Y∈D. This completes the proof.

Proof of Theorem 1.4. Let {ei,φei,ξ}i=1n be a local orthonormal basis as considered in Theorem 1.2. If the second order parallel tensor α is symmetric, then

Putting Y=Z=W=ξ in above, we get

Note that 2.6 takes the form

Using (3.15) in (3.14), and putting X=ei and φei successively in the resulting equation gives

and

from which we obtain

and

Since λi is different from +1 and -1, we get g(X,Aξ)=0 for any X⊥ξ. Hence Aξ=α(ξ,ξ)ξ. Now arguing in the similar manner as in Theorem 1.2, one can conclude that

for any X∈ TM, that is, α is a constant multiple of g. If α is skew-symmetric, then we have equation (3.9). Putting Y=Z=ξ and W=Aξ in (3.9), we find

Then using (3.15) in above, and putting X=e_i and φei successively we get g(X,Aξ)=0 for any X⊥ξ. Hence Aξ=α(ξ,ξ)ξ, and similar to the proof of Theorem 1.2, we obtain AX=0 for any X ∈ TM. This finishes the proof.

Proof of Theorem 1.5. Let {ei,φei,ξ}i=1n be a local orthonormal basis such that hei=λiei. The hypothesis trl>−2n−2 shows that ∑ i=1nλi2<1 which means λi2<1, that is, λi≠(+1,−1) for each i.

Now the condition £VR=0 implies

Let G be a (1,1)-tensor field defined by g(GX,Y)=(£Vg)(X,Y). Then

Taking Y=Z=W=ξ in (3.16), we get

Using (3.15) in above, and putting X=ei and φei successively in the resulting equation gives

and

from which we obtain

and

Since λi is different from +1 and -1, we get g(X,Gξ)=0 for any X⊥ξ. Hence Gξ=g(Gξ,ξ)ξ. Now putting Y=Z=ξ in 3.16, using 3.15, and taking X=e_i and φei successively, we obtain

and

So that, we have GX=g(Gξ,ξ)X for any X⊥ξ, and hence

that is,

for some function f. Thus V is a conformal vector field, and we have

Since £VR=0 implies £VS=0, the above equation gives ∇df=0, and hence

Since df⊗df is a (0,2)-tensor, it follows from Theorem 1.2 that df⊗df=cg, for some constant c. Thus

which for Y=gradf gives ‖gradf‖2gradf=c gradf. Now, if {ei} is an orthonormal basis then putting Y=ei in (3.17), taking inner product with e_i and summing over i yields

Consequently, we obtain f is constant and hence V is homothetic.