In 1982, Hamilton [12] introduced the notion of the Ricci flow to find a canonical metric on a smooth manifold. The Ricci flow is an evolution equation for metrics on a Riemannian manifold:

A Ricci soliton is a natural generalization of an Einstein metric and is defined on a Riemannian manifold (M, g). A Ricci soliton is a triple (g, V, λ) with g a Riemannian metric, V a vector field and λ a real scalar such that

where S is a Ricci tensor of M and L_V denotes the Lie derivative operator along the vector field V. The Ricci soliton is said to be shrinking, steady and expanding accordingly as λ is negative, zero and positive, respectively [10]. Ricci solitons have been studied in many contexts: on Kähler manifolds [11], on contact and Lorentzian manifolds [2, 14, 15, 17, 18], on Sasakian [13], α-Sasakian [1] and K-contact manifolds [19, 7], on Kenmotsu [3] and f-Kenmotsu manifolds [8] etc. In paracontact geometry, Ricci solitons firstly appeared in the paper of G. Calvaruso and D. Perrone [16]. Recently, C. L. Bejan and M. Crasmareanu dealed with Ricci solitons on 3-dimensional normal paracontact manifolds [4]. A more general notion is that of η-Ricci soliton introduced by J. T. Cho and M. Kimura [9], which was treated by C. Calin and M. Crasmareanu on Hopf hypersurfaces in complex space forms [8]. η-Ricci solitons on para-Kenmotsu manifolds were studied by A. M. Blaga [5] and η-Ricci solitons on Lorentzian Para-Sasakian Manifolds were also studied by A. M. Blaga [6]. Let (M, g), n =dimM ≥ 3, be a connected semi-Riemannian manifold of class C^∞ and ∇ be its Levi-Civita connection. The Riemannian-Christoffel curvature tensor R, the quasi-conformal curvature tensor C; the M-projective curvature tensor M̃; pseudo-projective curvature tensor P; the concircular curvature tensor C̃ and the conharmonic curvature tensor H of (M, g) are defined by

and

respectively, where Q is the Ricci operator, defined by S(X, Y) = g(QX, Y), S is the Ricci tensor, r = tr(S) is the scalar curvature and X, Y, Z ∈ χ(M), χ(M) being the Lie algebra of vector fields of M.

The paper is organized as follows:

In the present paper, we studied curvature properties of η-Ricci solitons on para-Kenmotsu manifolds. In section 2, we recall some well known basic formulas and properties of para-Kenmotsu manifolds. Section 3 contains a brief review of Ricci and η-Ricci solitons. In sections 4–8, we obtained some interesting results on η-Ricci solitons in para-Kenmotsu manifolds satisfying R(ξ, X).C = 0, R(ξ, X). M̃ = 0, R(ξ, X).P = 0, R(ξ, X). C̃ = 0 and R(ξ, X).H = 0, where C, M̃, P, C̃ and H are quasi-conformal curvature tensor ; M-projective curvature tensor; pseudo-projective curvature tensor; concircular curvature tensor and conharmonic curvature tensor, respectively.

Let (M, ϕ, η, ξ, g) be a n-dimensional smooth manifold, where ϕ is a tensor field of (1, 1)-type, η a 1-form, ξ a vector field and g a pseudo-Riemannian metric on M. We say that (ϕ, η, ξ, g) is an almost paracontact metric structure on M, if satisfies the conditions [5]:

for any vector fields X and Y on M.

If, moreover

where ∇ denotes the Levi-Civita connection of g, then the almost paracontact metric structure (ϕ, η, ξ, g) is called para-Kenmotsu manifold.

From the definition, it follows that η is the g-dual of ξ:

ξ is a unitary vector field:

and ϕ is a g-skew-symmetric operator.

We shall further give some immediate properties of this structure.

Proposition 2.1

On a para-Kenmotsu manifold (M, ϕ, η, ξ, g), the following relations hold:

where R is the Riemann curvature tensor field and ∇ is the Levi-Civita connection associated to g.

Let (M,ϕ, ξ, η, g) be a paracontact metric manifold. Consider the equation

where L_ξ is the Lie derivative operator along the vector field ξ, S is the Ricci curvature tensor field of the metric g, and λ and μ are real constants. Writing L_ξg in terms of the Levi-Civita connection ∇, we get

for any X, Y ∈ χ(M), or equivalent:

for any X, Y ∈ χ(M).

The data (g, ξ, λ, μ) which satisfy the equation (3.1) is said to be an η-Ricci soliton on M [8]; in particular, if μ = 0, (g, ξ, λ) is a Ricci soliton [18] and it is called shrinking, steady or expanding according as λ is negative, zero or positive, respectively [19].

Taking Y = ξ in (3.3), we get

On a n-dimensional paracontact manifold M, we have

so:

In this case, the Ricci operator Q defined by g(QX, Y) = S(X, Y) has the expression:

The above equation yields that

The Quasi-conformal curvature tensor C is defined by

where a, b ≠ 0 are constants. Putting Z = ξ in (4.1) and using (2.12), (3.3), (3.6), we obtain

Similarly using (2.13), (3.3), (3.4) and (3.6) in (4.1), we obtain

The condition that must be satisfied by R is:

By virtue of (2.11) and (4.4), we get

Taking the inner product with ξ, the relation (4.5) becomes:

By virtue of (4.2), (4.3) and (4.6), we get

By using (4.1) in (4.7) and putting X = U = e_i, summing over i = 1, 2, ..., n and on simplification, we have

Taking V = W = ξ in (4.8) and using (2.4), (3.7), we find the following equation

Thus, we can state the following theorem:

Theorem 4.1

If (ϕ, ξ, η, g) is a para-Kenmotsu structure on the n-dimensional manifold M, (ϕ, ξ, λ, μ) is an η-Ricci soliton on M and R(ξ, X).C = 0, then λ + μ − (n − 1) = 0 and (M, g) is an Einstein manifold.

The M-projective curvature tensor M̃ is defined by

Putting Z = ξ in (5.1) and using (2.12), (3.3), (3.6), we obtain

Similarly using (2.8), (3.3), (3.4), (3.6) in (5.1), we obtain

The condition that must be satisfied by R is:

By virtue of (2.11) and (5.4), we get

Taking the inner product with ξ, the relation (5.5) becomes:

By virtue (5.2), (5.3) and (5.6), we have

By using (5.1) in (5.7) and Putting X = Y = e_i, summing over i = 1, 2, ..., n and on simplification, we have

Taking V = W = ξ in (5.8) and by virtue of (3.6), (3.7), we find the following equation

Thus, we can state the following theorem:

Theorem 5.1

If (ϕ, ξ, η, g) is a para-Kenmotsu structure on the n-dimensional manifold M, (ϕ, ξ, λ, μ) is an η-Ricci soliton on M and R(ξ, X). M̃ = 0, then (2n− 1)λ + μ + 1 = 0 and (M, g) is an Einstein manifold.

The Pseudo-projective curvature tensor P is defined by

where a, b ≠ 0 are constants. Putting Z = ξ in (6.1) and using (2.12), (3.3), (3.6), we obtain

Similarly using (2.13), (3.3), (3.4), (3.6) in (6.1), we obtain

The condition that must be satisfied by R is:

By virtue of (2.11) and (6.4), we get

Taking the inner product with ξ, the relation (6.5) becomes:

By virtue of (6.2), (6.3) and (6.6), we have

By using (6.1) in (6.7) and Putting X = U = e_i, summing over i = 1, 2, ..., n and on simplification, we obtain

Taking V = W = ξ in (6.8) and by virtue of (3.4), (3.7), we find the following equation

Thus, we can state the following theorem:

Theorem 6.1

If (ϕ, ξ, η, g) is a para-Kenmotsu structure on the n-dimensional manifold M, (ϕ, ξ, λ, μ) is an η-Ricci soliton on M and R(ξ, X).P = 0, then λ + μ − (n − 1) = 0 and (M, g) is an η-Einstein manifold.

The concircular curvature tensor C̃ is defined by

Taking Z = ξ in (7.1) and using (2.12), (3.3), (3.6), we get

Similarly using (2.13), (3.3), (3.4), (3.6) in (7.1), we have

The condition that must be satisfied by R is:

By virtue of (2.11) and (7.4), we have

Taking the inner product with ξ, the relation (7.5) becomes:

By virtue of (7.2), (7.3) and (7.6), we get

By using (7.1) in (7.7) and Putting X = U = e_i, summing over i = 1, 2, ..., n and on simplification, we obtain

Taking V = W = ξ in (7.8) and by virtue of (3.4), (3.7), we find the following equation

Thus, we can state the following theorem:

Theorem 7.1

If (ϕ, ξ, η, g) is a para-Kenmotsu structure on the n-dimensional manifold M, (ϕ, ξ, λ, μ) is an η-Ricci soliton on M and R(ξ, X). C̃ = 0, then λ + μ − (n − 1) = 0 and (M, g) is an Einstein manifold.

The conharmonic curvature tensor H is defined by

Putting Z = ξ in (8.1) and using (2.12), (3.3), (3.6), we obtain

Similarly using (2.8), (2.13), (2.14), (3.5) in (8.1), we have

The condition that must be satisfied by R is:

By virtue of (2.11) and (8.4), we get

Taking the inner product with ξ, the relation (8.5) becomes:

By virtue of (8.2), (8.3) and (8.6), we get

By using (8.1) in (8.7) and Putting X = Y = e_i, summing over i = 1, 2, ..., n and on simplification, we obtain

where g(Z, W) ≠ 0. Therefore, we get

on simplification, we obtain

Thus, we can state the following theorem:

Theorem 8.1

If (ϕ, ξ, η, g) is a para-Kenmotsu structure on the n-dimensional manifold M, (ϕ, ξ, λ, μ) is an η-Ricci soliton on M and R(ξ, X).H = 0, then λ + μ − (n − 1) = 0.