Classification of Lagrangian submanifolds in complex space forms satisfying a basic equality involving δ(2,2)δ(2,2)

Lagrangian submanifolds appear naturally in the context of classical mechanics. They play important roles in geometry as well as in physics. It was proved by B.-Y. Chen in (2000) [6] that every Lagrangian submanifold M5M5 of a complex space form M˜5(4c) of constant holomorphic sectional curvature 4c satisfiesequation(A)δ(2,2)⩽253H2+8c, where H2H2 is the squared mean curvature and δ(2,2)δ(2,2) is a δ  -invariant of M5M5 (cf. Chen, 2000, 2011 [6] and [9]). The main purpose of this paper is to completely classify Lagrangian submanifolds of complex space forms M˜5(4c), c=0,1,−1c=0,1,−1, satisfying the equality case of the inequality (A) identically.