On the extrinsic principal directions and curvatures of Lagrangian submanifolds

TL;DR

This paper reviews the concepts of extrinsic principal directions and curvatures of submanifolds, focusing on their relationships in Lagrangian submanifolds within complex Euclidean spaces, highlighting historical and geometric insights.

Contribution

It explicitly characterizes the extrinsic principal directions and curvatures for Lagrangian submanifolds in complex Euclidean spaces, connecting classical and modern geometric concepts.

Findings

Relationships between principal directions and curvatures established

Explicit descriptions for Lagrangian submanifolds provided

Historical context of extrinsic directions summarized

Abstract

From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the $18$seventies), and what are the principal first normal directions, (first studied by Kostadin Trencevski in the $19$nineties), and what are their corresponding Casorati curvatures. For reasons of simplicity of exposition only, hereafter this will merely be done explicitly in the case of arbitrary submanifolds in Euclidean spaces. Then, for the special case of Lagrangian submanifolds in complex Euclidean spaces, the natural relationships between these distinguished tangential and normal directions and their corresponding curvatures will be established.