$\delta$-invariants for Lagrangian submanifolds of complex space forms

TL;DR

This paper reviews the development and application of $\, ext{ extdelta}$-invariants in studying Lagrangian submanifolds within complex space forms, highlighting their role in understanding extrinsic and intrinsic geometric properties.

Contribution

It compiles and discusses both historical and recent results on $\, extdelta$-invariants for Lagrangian submanifolds, emphasizing their significance in geometric analysis.

Findings

$\, extdelta$-invariants provide new tools for geometric inequalities.

Results connect $\, extdelta$-invariants with minimal immersion problems.

Advances improve understanding of extrinsic-intrinsic relations in complex space forms.

Abstract

The famous Nash embedding theorem published in 1956 was aiming for the opportunity to use extrinsic help in the study of (intrinsic) Riemannian geometry, if Riemannian manifolds could be regarded as Riemannian submanifolds. However, this hope had not been materialized yet according to \cite{G}. The main reason for this was the lack of control of the extrinsic properties of the submanifolds by the known intrinsic invariants. In order to overcome such difficulties as well as to provide answers to an open question on minimal immersions, the first author introduced in the early 1990's new types of Riemannian invariants, his so-called $\delta$-curvatures, different in nature from the "classical" Ricci and scalar curvatures. The purpose of this article is to present some old and recent results concerning $\delta$-invariants for Lagrangian submanifolds of complex space forms.