Journal of the
Korean Mathematical Society JKMS

Our aim is to study the properties of Fischer-Marsden conjecture and Ricci-Bourguignon solitons within the framework of generalized Sasakian-space-forms with $\beta$-Kenmotsu structure. It is proven that a $(2n+1)$-dimensional generalized Sasakian-space-form with $\beta$-Kenmotsu structure satisfying the Fischer-Marsden equation is a conformal gradient soliton. Also, it is shown that a generalized Sasakian-space-form with $\beta$-Kenmotsu structure admitting a gradient Ricci-Bourguignon soliton is either $\Psi \backslash T^{k} \times M^{2n+1-k}$ or gradient $\eta$-Yamabe soliton.

Keywords: Almost contact metric manifolds, generalized Sasakian-space-forms, Fischer-Marsden conjecture, Ricci-Bourguignon solitons, symmetric spaces

MSC numbers: Primary 53C25, 53C15

Supported by: The second author was supported by grant Proj.~No.~NRF-2018-R1D1A1B-05040381 from National Research Foundation of Korea.