Liftings from Lorentzian α-Sasakian manifolds to tangent bundles

The goal of the current study is to investigate the complete lift of Lorentzian α-Sasakian manifolds to the tangent bundle TM. We also examine the complete lift of the different four types of Lorentzian αSasakian manifolds and find that (TM,gC) is an η-Einstein manifold in each instance. In order to show that a Lorentzian α-Sasakian manifold exists on TM, a non-trivial example by means of partial differential equations is built in the final section.