By R and Ric we denote respectively the Riemannian curvature tensor and the Ricci tensor of a Riemannian manifold (M,g). Then R and Ric are defined by:

where ∇ is the Levi-Civita connection with respect to g, {e_i} is an orthonormal frame, and X, Y,Z ∈ Γ(TM). Given a smooth function f on M, the gradient of f is defined by:

the Hessian of f is defined by:

where X, Y ∈ Γ(TM). For X ∈ Γ(TM), we define X^♭ ∈ Γ(T^*M) by:

(For more details, see for example [11]).

- The generalised Ricci soliton equation in Riemannian manifold (M,g) is defined by (see [10]):

where X ∈ Γ(TM), ℒ_Xg is the Lie-derivative of g along X given by:

for all Y,Z ∈ Γ(TM), and c₁, c₂, λ ∈ ℝ. Equation (1.6), is a generalization of:

• Killing’s equation (c₁ = c₂ = λ = 0);

• Equation for homotheties (c₁ = c₂ = 0);

• Ricci soliton (c₁ = 0, c₂ = −1);

• Cases of Einstein-Weyl (c₁ = 1, c2=-1n-2);

• Metric projective structures with skew-symmetric Ricci tensor in projective class (c₁ = 1, c2=-1n-1, λ = 0);

• Vacuum near-horzion geometry equation (c₁ = 1, c2=12).

(For more details, see [5], [8], [9], [10]). Equation (1.6), is also a generalization of Einstein manifolds (see [1]). Note that, if X = grad f, where f ∈ C^∞(M), the generalised Ricci soliton equation is given by:

- An (2n + 1)-dimensional Riemannian manifold (M,g) is said to be an almost contact metric manifold if there exist a (1, 1)-tensor field ϕ on M, a vector field ξ ∈ Γ(TM) and a 1-form η ∈ Γ(T^*M), such that:

for all X, Y ∈ Γ(TM). In particular, in an almost contact metric manifold we also have ϕξ = 0, η ∘ ϕ = 0 and η(X) = g(ξ,X). It can be proved that an almost contact metric manifold is Sasakian if and only if (see [2], [3], [12]):

for any X, Y ∈ Γ(TM). For a Sasakian manifold the following equations hold:

The main result of this paper is the following:

Theorem 1.1

Suppose (M,ϕ, ξ, η, g) is a Sasakian manifold of dimension (2n+1), and satisfies the generalised Ricci soliton equation (1.8) with c₁(λ + 2c₂n) = −1, then f is a constant function. Furthermore, if c₂ = 0, then (M,g) is an Einstein manifold.

From Theorem 1.1, we get the following:

• Suppose (M,g) is a Sasakian manifold and satisfies the gradient Ricci soliton equation (i.e. Hess f = −Ric+λg), then f is a constant function and (M,g) is an Einstein manifold (this result is obtained by P. Nurowski and M. Randall [10]).

• In a Sasakian manifold (M,ϕ, ξ, η, g), there is no non-constant smooth function f, such that Hess f = λg, for some constant λ.

For the proof of Theorem 1.1, we need the following lemmas.

Lemma 2.1

Let (M,ϕ, ξ, η, g) be a Sasakian manifold. Then:

where X, Y ∈ Γ(TM), with Y is orthogonal to ξ.

Lemma 2.2

Let (M,g) be a Riemannian manifold, and let f ∈ C^∞(M). Then:

where ξ, Y ∈ Γ(TM).

Lemma 2.3

Let (M,ϕ, ξ, η, g) be a Sasakian manifold of dimension (2n+1), and satisfies the generalised Ricci soliton equation (1.8). Then:

The authors would like to thank the reviewers for their useful remarks and suggestions. Partially supported by National Agency Scientific Research of Algeria.