General Casorati inequalities and implications for Riemannian maps and Riemannian submersions

This paper presents general forms of Casorati inequalities for Riemannian maps and Riemannian submersions between Riemannian manifolds. Using these general forms, we obtain Casorati inequalities for Riemannian maps (resp. submersions) whose target (resp. source) spaces are generalized complex and generalized Sasakian space forms. As a consequence, we give Casorati inequalities for Riemannian maps (resp. submersions) when the target (resp. source) spaces are real, complex, real Kähler, Sasakian, Kenmotsu, cosymplectic, and almost $C(α)$ space forms. Toward the affirmation of these general forms, in the particular cases when the target or source spaces are real, complex, Sasakian, and Kenmotsu space forms, we verify known Casorati inequalities for Riemannian maps and Riemannian submersions. Further, we give Casorati inequalities for invariant and anti-invariant Riemannian maps (resp. submersions) whose target (resp. source) spaces are generalized complex and generalized Sasakian space forms. Toward information on geometric characteristics, we discuss the equality cases.