On properties of the Reeb vector field of $(\alpha,\beta)$ trans-Sasakian structure

Abstract

The paper focused on the mean curvature and totally geodesic property of the Reeb vector field $\xi$ on $(\alpha,\beta)$ trans-Sasakian manifold $M$ of dimension $(2n+1)$ as a submanifold in the unit tangent bundle $T_1M$ with Sasaki metric $g_S$. We give an explicit formula for the norm of mean curvature vector of the submanifold $\xi(M)\subset (T_1M,g_S)$. As a byproduct, for the Reeb vector field, we get some known results concerning its minimality, harmonicity and the property to define a harmonic map. We prove that on connected proper trans-Sasakian manifold the Reeb vector field does not give rise to totally geodesic submanifold in $T_1M$. On $\alpha$-Sasakian the Reeb vector field is totally geodesic only if $\alpha=1$. On $\beta$-Kenmotsu manifold the Reeb vector field is totally geodesic if and only if $\nabla\beta=\frac{\beta^2(1+\beta^2)}{1-\beta^2}\xi$. If $M$ is compact, then $\beta=0$.