Chen's inequality in the Lagrangian case

Abstract

Abstract. In the theory of submanifolds, the following problem is fundamental: establish simple relationships between the main intrinsic invariants and the main extrinsic invariants of submanifolds. The basic relationships discovered until now are inequalities.To analyze such problems, we follow the idea of C. Udrişte that the method of constrained extremum is a natural way to prove geometric inequalities. We improve 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.

Optimal general inequalities for Lagrangian submanifolds in complex space forms

“…Then we conclude that conditions (4.13)-(4.15) are equivalent to (4.2) due to the totally symmetry of h. 2 Remark 4.1. When k = 1 and n 1 = 2, inequality (4.1) is due to Oprea [15] (see also [13]). The equality case for this special case have been investigated rather detailed in [1][2][3].…”

Inequalities for casorati curvatures of submanifolds in real space forms

“…In [22], the above lemma was successfully applied to improve an inequality relating δ(2) obtained in [3]. Later, Chen extended the improved inequality to the general inequalities involving δ(n 1 , · · · , n k ) [4].…”

Lagrangian submanifolds attaining equality in the improved Chen's inequality

“…However, Oprea [7] has recently shown that the inequality (1) is not optimal, and, for n ≥ 3 can be improved to…”

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In [7] Oprea gave an improved version of Chen's inequality for Lagrangian submanifolds of CP n (4). For minimal submanifolds this inequality coincides with a previous version proved in [5].

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