General Chen-Ricci inequalities for Riemannian submersions and Riemannian maps

Abstract: We derive general forms of the Chen-Ricci inequalities for Riemannian submersions between Riemannian manifolds. We also derive general forms of the Chen-Ricci and improved Chen-Ricci inequalities for Riemannian maps between Riemannian manifolds, involving the relations between curvatures of subspaces of source and target spaces. These general forms give new techniques that are easy, elegant, and fruitful to obtain the Chen-Ricci inequalities for such smooth mappings with various space forms. As applications, utilizing these general forms, we explicitly establish Chen-Ricci inequalities when the source manifolds of Riemannian submersions and the target manifolds of Riemannian maps belong to broader classes, such as generalized complex and generalized Sasakian space forms, particularly including real, complex, real K\"ahler, Sasakian, Kenmotsu, cosymplectic, and $C(\alpha)$ space forms. Furthermore, we provide Chen-Ricci inequalities for various types of geometrically structured Riemannian submersions and Riemannian maps, such as invariant, anti-invariant, semi-invariant, slant, semi-slant, and hemi-slant. In particular, imposing suitable circumstances towards validation, our inequalities become the same as various known inequalities established by many authors in [1, 2, 6, 7, 9, 10, 22, 23, 29, 48, 49].