Ricci solitons in Lorentzian $\alpha$-Sasakian manifolds

Abstract: We study Ricci solitons in Lorentzian $\alpha$-Sasakian manifolds. It is shown that a symmetric parallel second order covariant tensor in a Lorentzian $\alpha$-Sasakian manifold is a constant multiple of the metric tensor. Using this it is shown that if $\mathcal L_Vg+2S$ is parallel, $V$ is a given vector field then $(g,V)$ is Ricci soliton. Further, by virtue of this result Ricci solitons for $(2n+1)$-dimensional Lorentzian $\alpha$-Sasakian manifolds are obtained. Next, Ricci solitons for 3-dimensional Lorentzian $\alpha$-Sasakian manifold whose scalar curvature is constant are obtained.

Keywords: Ricci soliton, Lorentzian metric, Sasakian manifold, Einstein

Classification (MSC2000): 53C15; 53C20, 53C21, 53C25, 53D10

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