Chen-Ricci inequalities for Riemannian maps and their applications

Abstract

Riemannian maps between Riemannian manifolds, originally introduced by A.E. Fischer in [Contemp. Math. 132 (1992), 331–366], provide an excellent tool for comparing the geometric structures of the source and target manifolds. Isometric immersions and Riemannian submersions are particular examples of such maps. In this work, we first prove a geometric inequality for Riemannian maps having a real space form as a target manifold. Applying it to the particular case of Riemannian submanifolds, we recover a classical result, obtained by B.-Y. Chen in [Glasgow Math. J. 41 (1999), 33–41], which nowadays is known as the Chen-Ricci inequality. Moreover, we extend this inequality in case of Riemannian maps with a complex space form as a target manifold. We also improve this inequality when the Riemannian map is Lagrangian. Applying it to Riemannian submanifolds, we recover the improved Chen-Ricci inequality for Lagrangian submanifolds in a complex space form, that is a basic inequality obtained by S. Deng in [Int. Electron. J. Geom. 2 (2009), 39-45] as an improvement of a geometric inequality stated by B.-Y. Chen in [Arch. Math. (Basel) 74 (2000), 154–160].