On quasi-Sasakian 3-manifolds admitting η-Ricci solitons
TL;DR: In this paper, it was shown that in a quasi-Sasakian 3-manifold admitting Ricci soliton, the structure function is a constant, which is the same as in the present paper.
Abstract: The object of the present paper is to prove that in a quasi-Sasakian
3-manifold admitting ?-Ricci soliton, the structure function ? is a
constant. As a consequence we obtain several important results.
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TL;DR: In this article , the existence of Ricci solitons in LP-Kenmotsu manifolds in the spacetime of general relativity has been proved through a nontrivial example.
Abstract: In the present paper, -Kenmotsu manifolds admitting -Ricci solitons have been studied. Moreover, some results for -Ricci solitons in LP-Kenmotsu manifolds in the spacetime of general relativity have also been proved. Through a nontrivial example, we have given a proof for the existence of η-Ricci solitons in a 5-dimensional LP-Kenmotsu manifold.
12 citations
TL;DR: In this paper, the authors characterize the gradient Yamabe and the gradient m-quasi Einstein solitons within the framework of three-dimensional cosymplectic manifolds.
Abstract: In this paper, we characterize the gradient Yamabe and the gradient m-quasi Einstein solitons within the framework of three-dimensional cosymplectic manifolds.
11 citations
TL;DR: In this article, it was shown that a 3D Riemannian manifold endowed with a semi-symmetric ρ-connection, whose metric is a Yamabe soliton, is a manifold of constant sectional curvature − 1 and the soliton is expanding with soliton constant − 6.
11 citations
TL;DR: In this article, the properties of perfect fluid spacetimes endowed with the gradient η -Ricci and gradient Einstein solitons were studied, and the authors set the goal to study the properties.
11 citations
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TL;DR: In this paper, a reduction result for parallel symmetric covariant tensor fields of order two was obtained for the parallel Ricci solitons with regularity and regularity conditions.
Abstract: Torse-forming $\eta $-Ricci solitons are studied in the framework of almost paracontact metric $\eta $-Einstein manifolds. By adding a technical condition, called regularity and concerning with the scalars provided by the two $\eta $-conditions, is obtained a reduction result for the parallel symmetric covariant tensor fields of order two.
28 citations
"On quasi-Sasakian 3-manifolds admit..." refers methods in this paper
...In this connection we may mention the works of Blaga ([7], [8], [9]), Prakasha et al....
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TL;DR: In this paper, a 3-dimensional quasi-Sasakian manifold is defined and the structure function β is defined on such a manifold, and necessary and sufficient conditions for M to be conformally flat are given.
Abstract: LetM be a 3-dimensional quasi-Sasakian manifold. On such a manifold, the so-called structure function β is defined. With the help of this function, we find necessary and sufficient conditions forM to be conformally flat. Next it is proved that ifM is additionally conformally flat with β = const., then (a)M is locally a product ofR and a 2-dimensional Kahlerian space of constant Gauss curvature (the cosymplectic case), or (b)M is of constant positive curvature (the non cosymplectic case; here the quasi-Sasakian structure is homothetic to a Sasakian structure). An example of a 3-dimensional quasi-Sasakian structure being conformally flat with nonconstant structure function is also described. For conformally flat quasi-Sasakian manifolds of higher dimensions see [O1]
23 citations
"On quasi-Sasakian 3-manifolds admit..." refers background or methods in this paper
...On a 3-dimensional quasi-Sasakian manifold, the structure function β was defined by Olszak [30] and with the help of this function he has obtained necessary and sufficient conditions for the manifold to be conformally flat [29]....
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...The Riemannian curvature tensor R of a 3-dimensional quasi-Sasakian manifold is given by [29]...
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...In [29], Olszak prove the following: Theorem 3....
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...[29] Let M is a quasi-Sasakian manifold of positive constant curvature K....
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...In a 3-dimensional quasi-Sasakian manifold, the Ricci tensor S is given by [29]...
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TL;DR: In this article, a Ricci solitons on a para-Sasakian manifold was studied and the results on the Ricci solution were given for the conditions 0 = ), ( R X S , 0 = ), ( P X R , and 0 = 0, where P and H are pseudo-projective and quasi-conharmonic curvature tensors respectively.
Abstract: The present paper aims at studying a Ricci solitons on para-Sasakian manifolds. We give the results on Ricci solitons in para-Sasakian manifold satisfying the conditions 0 = ) , ( R X S , 0 = ) , ( P X R and 0 = ) , ( H X R ,where P , H are pseudo-projective and quasi-conharmonic curvature tensors, respectively. MSC(2000): 53C21; 53C44; 53C25.
22 citations
01 Jan 1989
TL;DR: The authors trouve toutes les structures homogenes du groupe de Heisenberg generalise, and obtient une famille a 1 parametre de structures homogenees quasi sasakiennes sur ce groupe
Abstract: On trouve toutes les structures homogenes du groupe de Heisenberg generalise. On obtient une famille a 1 parametre de structures homogenes quasi sasakiennes sur ce groupe
21 citations
TL;DR: In this article, the divergence of the Ricci tensor of a metric connection with totally skew-symmetric torsion on a Riemannian manifold was studied.
Abstract: Let $
abla$ be a metric connection with totally skew-symmetric torsion $\T$ on a Riemannian manifold. Given a spinor field $\Psi$ and a dilaton function $\Phi$, the basic equations in type II B string theory are \bdm
abla \Psi = 0, \quad \delta(\T) = a \cdot \big(d \Phi \haken \T \big), \quad \T \cdot \Psi = b \cdot d \Phi \cdot \Psi + \mu \cdot \Psi . \edm We derive some relations between the length $||\T||^2$ of the torsion form, the scalar curvature of $
abla$, the dilaton function $\Phi$ and the parameters $a,b,\mu$. The main results deal with the divergence of the Ricci tensor $\Ric^{
abla}$ of the connection. In particular, if the supersymmetry $\Psi$ is non-trivial and if the conditions \bdm (d \Phi \haken \T) \haken \T = 0, \quad \delta^{
abla}(d \T) \cdot \Psi = 0 \edm hold, then the energy-momentum tensor is divergence-free. We show that the latter condition is satisfied in many examples constructed out of special geometries. A special case is $a = b$. Then the divergence of the energy-momentum tensor vanishes if and only if one condition $\delta^{
abla}(d \T) \cdot \Psi = 0$ holds. Strong models ($d \T = 0$) have this property, but there are examples with $\delta^{
abla}(d \T)
eq 0$ and $\delta^{
abla}(d \T) \cdot \Psi = 0$.
20 citations