On a class of $\beta$-Kenmotsu manifolds

Abstract

The object of the present paper is to study globally $\phi$-quasiconformally symmetric $\beta$-Kenmotsu manifolds. It has been shown that a globally $\phi$-quasiconformally symmetric $\beta$-Kenmotsu manifold is globally $\phi $-symmetric. Also we study $3$-dimensional locally $\phi $-symmetric $\beta$-Kenmotsu manifolds. next we study  second order parallel tensor and Ricci soliton on $3$-dimensional $\beta$-Kenmotsu manifolds. Finally, we give some examples of $3$-dimensional $\beta$-Kenmotsu manifolds which verifies our result.