Chen's inequality in Lagrangian case

Abstract

In the theory of submanifolds, the following problem is fundamental: to establish simple relationships between the main intrinsic invariants and the main extrinsic invariants of the submanifolds.The basic relationships discovered until now [1, 2, 3, 4] are inequalities. To analyze these problems, we follow the idea of C. Udriste [6, 7, 8] that the method of constrained extremum is a natural way to prove geometric inequalities. We improve the Chen's inequality which characterizes a totally real submanifold of a complex space form. For that we suppose that the submanifold is Lagrangian and we formulate and analyze a suitable constrained extremum problem.