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Abstract In this paper, using optimization methods on Riemannian submanifolds, we establish two improved inequalities for generalized normalized δ -Casorati curvatures of Lagrangian submanifolds in complex space forms. We provide examples showing that these inequalities are the best possible and classify all Casorati ideal Lagrangian submanifolds (in the sense of B.-Y. Chen) in a complex space form. In particular, we generalize the recent results obtained in G.E. Vîlcu (2018) [34]. [ABSTRACT FROM AUTHOR]

Abstract: Let be an n-dimensional Lagrangian submanifold of a complex space form of constant holomorphic sectional curvature 4c. We prove a pointwise inequality with on the left-hand side any delta-invariant of the Riemannian manifold and on the right-hand side a linear combination of the squared mean curvature of the immersion and the constant holomorphic sectional curvature of the ambient space. The coefficients on the right-hand side are optimal in the sense that there exist non-minimal examples satisfying equality at least one point. We also characterize those Lagrangian submanifolds satisfying equality at any of their points. Our results correct and extend those given in [6]. [Copyright &y& Elsevier]

Abstract: 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) that every Lagrangian submanifold of a complex space form of constant holomorphic sectional curvature 4c satisfies where is the squared mean curvature and is a δ-invariant of (cf. Chen, 2000, 2011 ). The main purpose of this paper is to completely classify Lagrangian submanifolds of complex space forms , , satisfying the equality case of the inequality (A) identically. [Copyright &y& Elsevier]

Abstract: We present Chen–Ricci inequality and improved Chen–Ricci inequality for curvature like tensors. Applying our improved Chen–Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms, and C-totally real submanifolds of Sasakian space forms. [Copyright &y& Elsevier]

Abstract: A local characterization of flat affine Lagrangian surfaces in is given. Metrizability of such surfaces is discussed. [Copyright &y& Elsevier]

Abstract: In this work we study pseudo-parallel Lagrangian submanifolds in a complex space form. We give several general properties of pseudo-parallel submanifolds. For the 2-dimensional case, we show that any minimal Lagrangian surface is pseudo-parallel. We also give examples of non-minimal pseudo-parallel Lagrangian surfaces. Here we prove a local classification of the pseudo-parallel Lagrangian surfaces. In particular, semi-parallel Lagrangian surfaces are totally geodesic or flat. Finally, we give examples of pseudo-parallel Lagrangian surfaces which are not semi-parallel. [Copyright &y& Elsevier]

Abstract: We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of such surfaces. We also use the transforms to show how to associate to such a surface a corresponding ruled minimal Lagrangian submanifold of complex projective 3-space, which gives the converse of a construction considered in a previous paper, and illustrate this explicitly in the case of bipolar minimal surfaces. [Copyright &y& Elsevier]

Abstract: We uncover the lowest order differential invariants of Lagrangian submanifolds under affine symplectic maps, and find out what happens when they are constant. [Copyright &y& Elsevier]

Abstract: We prove a conjecture formulated by Pablo M. Chacon and Guillermo A. Lobos in [P.M. Chacon, G.A. Lobos, Pseudo-parallel Lagrangian submanifolds in complex space forms, Differential Geom. Appl. 27 (1) (2009) 137–145, doi:10.1016/j.difgeo.2008.06.014] stating that every Lagrangian pseudo-parallel submanifold of a complex space form of dimension at least 3 is semi-parallel. We also propose to study another notion of pseudo-parallelity which is more adapted to the Kaehlerian setting. [Copyright &y& Elsevier]

In this paper a bijective correspondence between superminimal surfaces of an oriented Riemannian 4-manifold and particular Lagrangian submanifolds of the twistor space over the 4-manifold is proven. More explicitly, for every superminimal surface a submanifold of the twistor space is constructed which is Lagrangian for all the natural almost Hermitian structures on the twistor space. The twistor fibration restricted to the constructed Lagrangian gives a circle bundle over the superminimal surface. Conversely, if a submanifold of the twistor space is Lagrangian for all the natural almost Hermitian structures, then the Lagrangian projects to a superminimal surface and is contained in the Lagrangian constructed from this surface. In particular this produces many Lagrangian submanifolds of the twistor spaces C P 3 and F 1 , 2 (C 3) with respect to both the Kähler structure as well as the nearly Kähler structure. Moreover, it is shown that these Lagrangian submanifolds are minimal submanifolds. [ABSTRACT FROM AUTHOR]