Chen-type inequalities for Riemannian submersions

This study as a philosophy doctoral thesis covers four main chapters. In the first chapter, the historical developments and the problems which are discussed in the thesis are presented. Then, the basic definitions and theorems on Riemannian manifolds are expressed and general facts about Riemannian submersions are given. Moreover, some basic notions are given on Hermitian submersions between Hermitian manifolds. In particular, we focus on a Lagrangian Riemannian submersion from Kaehler manifold to Riemannian manifold and their properties are examined. In the second chapter, some inequalities are established involving curvature invariants such as sectional curvature, scalar curvature and mean curvature for Riemannian submersions between Riemannian manifolds. Using these inequalities, some characterizations are obtained. Also, Chen-Ricci inequality is established for such a submersion and some basic examples are presented which is satisfied Chen-Ricci inequality. On the other hand, some inequalities are given for Lagrangian Riemannian submersions using curvature invariants and the equality case of these inequalities are considered. Moreover, the harmonicity of Lagrangian Riemannian submersion is studied and the necessary and sufficient conditions for which such a submersion is harmonic are provided. In the third chapter, the results are presented which are obtained from this philosophy doctoral thesis and some suggestions on this topic are given in the last chapter.