On the vertical strong sphericity of Sasaki metric of tangent sphere bundles

The distribution $\mathcal L^q$ on the Riemannian manifold $M^n$ is called strong spherical if the curvature tensor of its metric satisfies the condition $R(X,Y)Z=k(\langle Y,Z\rangle X-\langle X,Z\rangle Y)$, ($k>0$) for any tangent to $M^n$ vectors $X$, $Z$ and any $Y\in\mathcal L^q$. The value $q=\operatorname{dim}\mathcal L^q$ is called the strong sphericity index. The conditions are considered at winch the vertical strong spherical distribution can exist on tangent sphere bundle $T_1M^n$ with Sasaki metric.