On the Extrinsic Principal Directions and Curvatures of Lagrangian Submanifolds

Marilena Moruz,Leopold Verstraelen

doi:10.3390/math8091533

Abstract

From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the 18seventies), and what are the principal first normal directions, (first studied by Kostadin Trenčevski in the 19nineties), 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.

Highlights

For general submanifolds Mn of dimension n (≥ 2) and of co-dimension m (≥ 1) in Euclidean spaces En+m, Jordan [1] studied the extrinsic curvatures cuT ( p) at arbitrary points p ∈ M in arbitrary tangential directions determined by vectors u ∈ Tp M, kuk = 1
In the first step of his original fundamental studies of the geometry of submanifolds, Trencevski [2,3,4,5] re-considered this work of Jordan, and, later, Stefan Haesen and Daniel Kowalczyk and one of the authors [6] basically re-did this
To continue in our aim to aim for simplicity and concreteness of presentation, we restrict our attention to the real n dimensional totally real submanifolds Mn of the complex n dimensional complex Euclidean spaces Mn = Cn = (E2n, J ), that is, to the Lagrangian submanifolds Mn in Cn, having J ( TM ) = T ⊥ M and J ( T ⊥ M) = TM, Jbeing the complex structure of the Kaehler manifold Mn

Summary

The Extrinsic Tangential Principal Directions of Submanifolds

For general submanifolds Mn of dimension n (≥ 2) and of co-dimension m (≥ 1) in Euclidean spaces En+m , Jordan [1] studied the extrinsic curvatures cuT ( p) at arbitrary points p ∈ M in arbitrary tangential directions determined by vectors u ∈ Tp M, kuk = 1. These are the curvatures cuT ( p) =.

Felice Casorati’s Study of Surfaces M 2 in E3

The First Normal Principal Directions of Submanifolds