For unit tangent sphere bundles $T_1M$ with the standard contact metric structure (${eta},ar{g},{phi},{xi}$), we have two fundamental operators that is, $h=frac{1}{2}{pounds}_{xi}{phi}$ and ${ell}=ar{R}({cdot},{xi}){xi}$, where ${pounds}_{xi}$ denotes Lie differentiation for the Reeb vector field ${xi}$ and $ar{R}$ denotes the Riemmannian curvature tensor of $T_1M$. In this paper, we study the Reeb ow invariancy of the corresponding (0, 2)-tensor fields H and L of h and ${ell}$, respectively.