Characterizations of three-dimensional trans-Sasakian generalized Sasakian space-forms admitting ⋆-Fischer-Marsden equation and Schouten solitons

Abstract

In the present article, we define the ⋆-Fischer–Marsden equation on three-dimensional trans-Sasakian generalized Sasakian space-forms. We show that if a three-dimensional trans-Sasakian generalized Sasakian space-form admits a non-trivial solution of the ⋆-Fischer–Marsden equation, then the structure is β-Kenmotsu. We also investigate Schouten solitons on three-dimensional trans-Sasakian generalized Sasakian space-forms. We prove that if the metric of a trans-Sasakian generalized Sasakian space-form is a gradient Schouten soliton, then the manifold is η-Einstein. An example is constructed to illustrate our results.