After being introduced by Kaneyuki and Williams [10] in 1985, a systematic study of paracontact metric manifolds and their subclasses, especially para-Sasakian manifolds, was carried out by Zamkovoy [21]. Paracontact metric manifolds have been studied by several authors such as Alekseevski et. al. [1, 2], Cortés [6], Erdem [9], Martin-Molina [12]. Recently, Cappelletti-Montano et. al. [5] introduced a new type of paracontact geometry, so-called paracontact metric (k, μ)-spaces, where k and μ are real constants. It is well known [3] that in the contact case one requires k ≤ 1, but there is no such restriction for k in the paracontact case [5]. Also, in the contact case, k = 1 implies the manifold is Sasakian but in paracontact case, k = −1 does not imply the manifold is para-Sasakian.

Among the geometric properties of manifolds symmetry is an important one. From the local point of view it was introduced by Shirokov [18] as a Riemannian manifold with covariant constant curvature tensor R, that is, with ∇R = 0, where ∇ is the Levi-Civita connection. An extensive theory of symmetric Riemannian manifolds was carried out by Cartan in 1927. A manifold is called semisymmetric if the curvature tensor R satisfies R(X, Y) · R = 0, where R(X, Y) is considered to be a derivation of the tensor algebra at each point of the manifold for the tangent vectors X, Y. Semisymmetric manifolds were locally classified by Szabó [19]. A manifold is said to be Ricci semisymmetric if R(X, Y) · S = 0 where S denotes the Ricci tensor of type (0, 2). A general classification of these manifolds has been worked out by Mirzoyan [13].

The notion of locally φ-symmetric was introduced by Takahashi [20] in Sasakian geometry as a weaker version of locally symmetric manifolds. In [7], De and Sarkar studied φ-Ricci symmetric Sasakian manifolds. Prakasha and Mirji [15] studied φ-Ricci symmetric N(k)-paracontact metric manifolds.

Also one of the main purposes of this paper is to study Eisenhart problem. In 1923, Eisenhart [8] proved that if a positive definite Riemannian manifold admits a second order parallel symmetric covariant tensor other than a constant multiple of the associated metric tensor, then it is reducible. In 1925, Levy [11] proved that a second order symmetric parallel non-singular tensor on a space of constant curvature is a constant multiple of the associated metric tensor. Sharma [16, 17] extended the result in contact geometry. Recently, Mondal et. al. [14] studied second order parallel tensors on (k, μ)-contact metric manifolds. Here we consider second order parallel symmetric covariant tensors on paracontact metric (k, μ)-spaces.

A paracontact metric (k, μ)-manifold is said to be an Einstein manifold if the Ricci tensor satisfies S = λg, where λ is some constant.

The paper is organized as follows:

In Section 2, we provide some basic results of paracontact metric (k, μ)-manifolds. Sections 3 and 4 are devoted to study Ricci semisymmetric and φ-Ricci symmetric paracontact metric (k, μ)-manifolds, respectively. In Section 5, we study the existence of symmetric parallel covariant tensors on paracontact metric (k, μ)-spaces. Several consequences of these results are discussed.

A smooth manifold M²ⁿ⁺¹ is said to admit an almost paracontact structure (φ, ξ, η) if it admits a tensor field φ of type (1, 1), a vector field ξ and a 1-form η satisfying [10]

• φ²X = X − η(X)ξ, for any vector field X ∈ χ(M), the set of all differential vector fields on M,

• φ(ξ) = 0, η ∘ φ = 0, η(ξ) = 1,

• the tensor field φ induces an almost paracomplex structure on each fibre of , that is, the eigen distributions Dφ+ and Dφ- of φ corresponding to the eigenvalues 1 and −1, respectively, have same dimension n.

An almost paracontact structure is said to be normal [21] if and only if the (1, 2)-type torsion tensor N_φ = [φ, φ] − 2dη ⊗ ξ vanishes identically, where [φ, φ](X, Y) = φ²[X, Y] + [φX, φY] − φ[φX, Y] − φ[X, φY]. An almost paracontact manifold equipped with a pseudo-Riemannian metric g such that

for all X, Y ∈ χ(M), is called almost paracontact metric manifold, where signature of g is (n + 1, n). An almost paracontact structure is said to be a paracontact structure if g(X, φY) = dη(X, Y) with the associated metric g [21]. For any almost paracontact metric manifold (M²ⁿ⁺¹, φ, ξ, η, g) admits (at least, locally) a φ-basis [21], that is, a pseudo-orthonormal basis of vector fields of the form {ξ, E₁, E₂, ..., E_n, φE₁, φE₂, ...,φE_n}, where ξ, E₁, E₂, ..., E_n are space-like vector fields and then, by (2.1) the vector fields φE₁, φE₂, ...,φE_n are time-like. In a paracontact metric manifold we define a symmetric, trace-free (1, 1)-tensor h=12ℒξφ satisfying [21]

where ∇ is the Levi-Civita connection of the pseudo-Riemannian manifold. Noticing that the tensor h vanishes identically if and only if ξ is a Killing vector field and in such case (φ, ξ, η, g) is said to be a K-paracontact structure. An almost paracontact manifold is said to be para-Sasakian if and only if the following condition holds [21]

for any X, Y ∈ χ(M). A normal paracontact metric manifold is para-Sasakian and satisfies

for any X, Y ∈ χ(M), but unlike contact metric geometry (2.5) is not a sufficient condition for a paracontact manifold to be para-Sasakian. It is clear that every para-Sasakian manifold is K-paracontact, but the converse is not always true, as it is shown in three dimensional case [4].

Finally, we recall the definition of paracontact metric (k, μ)-manifolds [5]:

Definition 2.1

A paracontact metric manifold is said to be a paracontact (k, μ)-manifold if the curvature tensor R satisfies

for all vector fields X, Y ∈ χ(M) and k, μ are real constants.

This class is very wide containing the para-Sasakian manifolds [10, 21] as well as the paracontact metric manifolds satisfying R(X, Y)ξ = 0 for all X, Y ∈ χ(M) [22].

In particular, if μ = 0, then the paracontact metric (k, μ)-manifold is called paracontact metric N(k)-manifold. Thus for a paracontact metric N(k)-manifold the curvature tensor satisfies the following relation

for all X, Y ∈ χ(M). Though the geometric behavior of paracontact metric (k, μ)-spaces is different according as k < −1, or k > −1, or k = −1, but there are some common results for k < −1 and k > −1. In [5], Cappelletti-Montano et. al. pointed out the following result.

Lemma 2.1. ([5], p.686, 692)

There does not exist any paracontact (k, μ)-manifold of dimension greater than 3 with k > −1 which is Einstein whereas there exists such manifolds for k < −1.

In a paracontact metric (k, μ)-manifold (M²ⁿ⁺¹, φ, ξ, η, g), n > 1, the following relations hold [5]:

for any vector fields X, Y ∈ χ(M), where Q is the Ricci operator defined by g(QX, Y) = S(X, Y). Making use of (2.3) we have

for all vector fields X, Y ∈ χ(M).

According to Takahashi [20] we have the following:

Definition 2.2

A paracontact metric (k, μ)-manifold is said to be φ-symmetric if it satisfies

for any vector fields W, X, Y and Z ∈ χ(M). In addition, if the vector fields W, X, Y, Z are horizontal then the manifold is called locally φ-symmetric. It is to be noted that φ-symmetry implies locally φ-symmetry, but the converse is not true, in general.

In this section we discuss about Ricci semisymmetric paracontact metric (k, μ)-manifolds. Suppose the paracontact metric (k, μ)-manifold M be Ricci semisymmetric. Then

for all X, Y ∈ χ(M). This is equivalent to

for any U, V, X, Y ∈ χ(M). Thus we have

Substituting X = U = ξ in (3.2) yields

Using (2.11) we infer from (3.3)

From (2.6) it follows that

With the help of (3.4) and (3.5) we get

Putting Y = hY in (3.6) and using (2.8) we obtain

Now suppose k < −1 and μ ≠ 0. Multiplying Equation (3.6) by k and Equation (3.7) by μ, then subtract the results we get

If k < −1, then k² − μ²(k + 1) ≠ 0. Therefore from (3.8) it follows that S(Y, V) = 2nkg(Y, V), which implies that the manifold is Einstein.

Also, if we take k < −1 and μ = 0, then (3.6) becomes

This implies S(Y, V) = 2nkg(Y, V), that is, the manifold is Einstein one.

Conversely, if the manifold is an Einstein manifold, then it can be easily shown that R · S = 0.

This leads to the following:

Theorem 3.1

A (2n+1)-dimensional (n > 1) paracontact metric (k, μ)-manifold with k < −1 is Ricci semisymmetric if and only if the manifold is Einstein.

Again Ricci symmetry (∇S = 0) implies Ricci semisymmetric (R · S = 0), therefore we have the following:

Corollary 3.1

A (2n+1)-dimensional (n > 1) paracontact metric (k, μ)-manifold with k < −1 is Ricci symmetric if and only if the manifold is Einstein.

Taking covariant derivative of (2.10) along an arbitrary vector field X, we have

Using (2.13) in the above equation gives

Thus the condition (∇_XQ)Y = 0 holds if and only if k=1n-n and μ = −2(n−1). Hence we can state the following:

Corollary 3.2

A (2n+1)-dimensional (n > 1) paracontact metric (k, μ)-manifold is Ricci symmetry (∇S = 0) if and only ifk=1n-nand μ = −2(n − 1).

Together with Corollary 5.12 of [5] we have the following:

Corollary 3.3

A (2n+1)-dimensional (n > 1) paracontact metric (k, μ)-manifold with k < −1 is Ricci symmetry (∇S = 0) if and only if the manifold is Einstein.

Since semisymmetry (R · R = 0) implies Ricci semisymmetry (R · S = 0), we can state the following:

Corollary 3.4

A (2n+1)-dimensional (n > 1) paracontact metric (k, μ)-manifold with k < −1 is semisymmetric if and only if the manifold is Einstein.

Remark 3.1

If k > −1, then from Lemma 2.1 and Equation (3.8) yields k² − μ²(k + 1) = 0.

In this section we characterize φ-Ricci symmetric paracontact metric (k, μ)-manifolds.

Definition 4.1.([7])

A paracontact metric (k, μ)-manifold is said to be φ-Ricci symmetric if it satisfies

for any vector fields X, Y ∈ χ(M). The manifold is called locally φ-Ricci symmetric if (4.1) holds for any horizontal vector fields. It follows that φ-Ricci symmetry implies locally φ-Ricci symmetry, but the converse is not true.

Let M be a (2n + 1)-dimensional (n > 1) paracontact metric (k, μ)-manifold. From (4.1) we have

for any vector fields X, Y ∈ χ(M).

Taking inner product of (4.2) with arbitrary vector field Z we obtain

This implies

Substituting Y = ξ in (4.4) gives

Taking covariant derivative of (2.10) along arbitrary vector field X, we obtain

Also from (2.13) we get

Making use of (2.15), (4.7) and (4.6) one can easily obtain

Taking account of (2.3), (4.8) and (4.5) we have

Replacing X by hX in (4.9) and using (2.8) gives that

Adding (4.9) and (4.10) we obtain

Since k ≠ 0, (4.11) implies

Putting X = φX in (4.12) yields

which shows that the manifold is an Einstein manifold.

Conversely, suppose S(X, Z) = 2nkg(X, Z), which implies QX = 2nkX. Hence (∇_Y Q)X = 0, that is, φ²((∇_Y Q)X) = 0. Therefore the manifold is φ-Ricci symmetric. Thus we can state the following.

Theorem 4.1

A (2n+1)-dimensional (n > 1) paracontact metric (k, μ)-manifold is φ-Ricci symmetric if and only if the manifold is an Einstein manifold, provided k ≠ 0,−1.

By the above arguments together with μ = 0 we have the following:

Corollary 4.1

A (2n+1)-dimensional (n > 1) paracontact metric N(k)-manifold is φ-Ricci symmetric if and only if the manifold is an Einstein manifold, provided k ≠ 0,−1.

Taking covariant differentiation of (2.10) along an arbitrary vector field X, we obtain

Using (2.13) in the above equation, we get

Applying φ² on both sides of (4.15) and making use of (2.8) gives

This is equivalent to

From the foregoing equation we see that φ²((∇_XQ)Y) = 0 if and only if k=1n-n and μ = −2(n − 1). This leads to the following:

Theorem 4.2

A (2n+1)-dimensional (n > 1) paracontact metric (k, μ)-manifold is φ-Ricci symmetric if and only ifk=1n-nand μ = −2(n − 1).

Hence from the Corollary 5.12 of [5] we conclude the following:

Corollary 4.2

A (2n+1)-dimensional (n > 1) paracontact metric (k, μ)-manifold with k < −1 is φ-Ricci symmetric if and only if the manifold is Einstein.

Definition 5.1. ([11])

A tensor α of second order is said to be parallel if ∇α = 0, where ∇ denotes the covariant differentiation with respect to the associated metric tensor.

Let α be a symmetric (0, 2)-tensor field on a paracontact metric (k, μ)-manifold M such that ∇α = 0. Then it follows that

for any vector fields X, Y, Z, W ∈ χ(M).

Substituting X = Z = W = ξ in (5.1) and noticing α is symmetric implies

Now we consider a non-empty connected open subset of M and restrict our discussions to this set. Applying (2.6) in (5.2) yields

We now consider the following cases:

• Case 1. k < −1, μ = 0,

• Case 2. k < −1, μ ≠ 0.

For the Case 1, we have from (5.3)

Taking covariant differentiation of (5.4) along X, we obtain

Replacing Y by ∇_XY in (5.4), we get

Using (5.5) and (5.6) we have

Making use of (2.3) and (5.4) in (5.7) follows that

Changing X by φX in (5.8) and using (2.2) we have

Putting X = hX in (5.9) and making use of (2.8) we obtain

Subtracting (5.9) from (5.10) and since k ≠ 0 it follows that

Since α and g are parallel tensor fields, α(ξ, ξ) must be constant on . Since is an arbitrary open set of M, it follows that (5.11) holds on whole of M.

For Case 2, replacing Y by hY in (5.3) and using (2.8) we have

Multiplying Equation (5.3) by k and Equation (5.12) by μ (since k < −1 and μ ≠ 0), then adding the results we get

Since k < −1, we see that k² − μ²(k + 1) ≠ 0. Hence, it follows from (5.13) that the relation (5.4) holds and then proceeding in the same way as in Case 1, we can show that α(X, Y) = α(ξ, ξ)g(X, Y) for all X, Y ∈ χ(M).

Considering the above facts we can state the following:

Theorem 5.1

Let M be a (2n+1)-dimensional (n > 1) paracontact metric (k, μ)-manifold with k < −1. If M admits a second order symmetric parallel tensor then it is a constant multiple of the associated metric tensor.

Corollary 5.1

A (2n + 1)-dimensional (n > 1) Ricci symmetric (∇S = 0) paracontact metric (k, μ)-manifold with k < −1 is an Einstein manifold.

The authors are grateful to the Editor-in-Chief for careful reading of the paper and valuable suggestions and comments towards the improvement of the paper.