A tour through $\delta$-invariants: From Nash's embedding theorem to ideal immersions, best ways of living and beyond

TL;DR

This paper introduces $\,delta$-invariants for Riemannian manifolds, explores their applications, and presents recent optimal inequalities for Lagrangian submanifolds, connecting geometric concepts with broader mathematical contexts.

Contribution

It introduces $\,delta$-invariants, discusses ideal immersions and optimal inequalities, and applies these concepts to various mathematical areas.

Findings

Introduction of $\,delta$-invariants for Riemannian manifolds

Development of two optimal inequalities for Lagrangian submanifolds

Applications of $\,delta$-invariants across multiple mathematical fields

Abstract

First I will explain my motivation to introduce the $\delta$-invariants for Riemannian manifolds. I will also recall the notions of ideal immersions and best ways of living. Then I will present a few of the many applications of $\delta$-invariants to several areas in mathematics. Finally, I will present two optimal inequalities involving $\delta$-invariants for Lagrangian submanifolds obtained very recently in joint papers with F. Dillen, J. Van der Veken and L. Vrancken.