Optimal inequalities involving Casorati curvatures along Riemannian maps and Riemannian submersions for Sasakian space forms

In this paper, Casorati inequalities are obtained for Riemannian maps and Riemannian submersions defined on Sasakian manifolds, and geometric results are given for the equality cases. First, Casorati inequalities for a Riemannian map from a Sasakian space form to a Riemannian manifold are obtained, and the equality case holds from geometric properties. Afterwards, Casorati inequalities involving tensor fields T and A are obtained for a Riemannian submersion from a Sasakian space form to a Riemann manifold, and geometric interpretations are given. It is shown that the equality of the inequalities obtained for tensor field A is equivalent to the integrability of the horizontal distribution. In the last section, Casorati inequalities and geometric results of a Riemannian map from a Sasakian manifold to a Sasakian space form are given.