A study on Ricci solitons and gradient Ricci solitons in three-dimensional trans sasakian manifoldes

In this study, the article on Ricci solitons and gradient Ricci solitons in 3-dimensional trans-Sasakian manifolds , written by Turan, M., De, U.C. ve Yıldız, A.(2012), were investigated. A Ricci soliton is the general form of the Einstein metric. In the manifold (M, g) satisfies £ vg + 2S + 2λg = 0 (*) then g is called a Ricci Soliton. £ Lie derivative, S is a Ricci tensor, V is a complete vector field on M and λ is constant. The metrics that satisfies the equation (*) are interesting and very useful in physics. Ricci soliton is said to be narrowed, stable and expanding depending on whether λ is negative, zero and positive respectively. If the vector field V is the gradient of the potential function -f , then g is called gradient Ricci soliton. In this case (*) equation takes following form: ∇∇ f = S + λg If the metric g in a 3-dimensional trans-Sasakian manifold Ricci soliton linear with the characteristic vector V and α, β is constant, then the manifold has constant-scalar curvature. If a 3-dimensional trans-Sasakian manifold has constant scalar curvature, then this manifold is either β-Kenmotsu manifold or Einstein manifold. In order to be understood more easily, some basic concepts were included in the study. But at the beginning of the study, a brief history of the concept of the manifold, which is one of the fundamental theories of differential geometry, was described for enthusiasts. In addition, tensor, Ricci tensor, soliton function concepts were also included.2019, 83 pages Keywords: 3-dimensional trans-Sasakian manifolds, Ricci Tensor, Ricci Soliton