Kyungpook Mathematical Journal 2017; 57(4): 677-682
Published online December 23, 2017
Copyright © Kyungpook Mathematical Journal.
Generalised Ricci Solitons on Sasakian Manifolds
Mohammed El Amine Mekki and Ahmed Mohammed Cherif
Department of Mathematics, University Mustapha Stambouli, Mascara, 29000, Algeria
Received: January 12, 2017; Accepted: July 18, 2017
In this paper, we show that a Sasakian manifold which also satisfies the generalised gradient Ricci soliton equation, satisfying some conditions, is necessarily Einstein.
Keywords: Sasakian manifolds, Ricci soliton, Einstein manifolds
1. Introduction and Main Results
where ∇ is the Levi-Civita connection with respect to
the Hessian of
(For more details, see for example ).
- The generalised Ricci soliton equation in Riemannian manifold (
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, );
Metric projective structures with skew-symmetric Ricci tensor in projective class (
c1 = 1, , λ = 0);
Vacuum near-horzion geometry equation (
c1 = 1, ).
(For more details, see , , , ).
- An (2
The main result of this paper is the following:
From Theorem 1.1, we get the following:
M,g) is a Sasakian manifold and satisfies the gradient Ricci soliton equation (i.e. Hess f= −Ric+λ g), then fis a constant function and ( M,g) is an Einstein manifold (this result is obtained by P. Nurowski and M. Randall ).
In a Sasakian manifold (
M,ϕ, ξ, η, g), there is no non-constant smooth function f, such that Hess f= λ g, for some constant λ.
2. Proof of the Result
For the proof of Theorem 1.1, we need the following lemmas.
First note that:
by the definition of the Riemannian curvature tensor (
the Lemma follows from
The proof is completed.
the Lemma follows from
by Lemma 2.3, and
from Lemma 2.3,
Note that, from
is equivalent to:
according to Lemma 2.3, we have:
The authors would like to thank the reviewers for their useful remarks and suggestions. Partially supported by National Agency Scientific Research of Algeria.
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