For unit tangent sphere bundles $T_1 M$ with the standard contact metric structure $(\eta,\bar g,\phi,\xi)$, we have two fundamental operators that is, $h=\frac{1}{2} \pounds_\xi\phi$ and $\ell=\bar R(\cdot,\xi)\xi$, where $\pounds_\xi$ denotes Lie differentiation for the Reeb vector field $\xi$ and $\bar R$ denotes the Riemmannian curvature tensor of $T_1 M$. In this paper, we study the Reeb flow invariancy of the corresponding $(0,2)$-tensor fields $H$ and $L$ of $h$ and $\ell$, respectively.