Exemples de sous vari\'et\'es de $\mathbb{S}^n$; les vari\'et\'es proprement biharmoniques avec indice et les vari\'et\'es proprement C--harmoniques

TL;DR

This paper investigates specific submanifolds of the sphere that are proper indexed-biharmonic, generalizing harmonic maps, and explores their properties and relationships with conformal-harmonic submanifolds.

Contribution

It introduces the concept of indexed-biharmonic submanifolds of spheres and analyzes their geometric properties and connections to conformal-harmonic submanifolds.

Findings

Constructed pseudo-umbilical proper indexed-biharmonic examples

Analyzed non pseudo-umbilical proper indexed-biharmonic submanifolds

Established links between indexed-biharmonic and conformal-harmonic submanifolds

Abstract

We call indexed-biharmonic maps, the solutions of a particular non linear elliptic PDE of order 4. This is a generalization of harmonic maps which verifies that biharmonic maps are biharmonic of index 0. The goal of this article is to study submanifolds of $\mathbb{S}^n$ whose inclusion is non harmonic and indexed-biharmonic (i.e. proper indexed-biharmonic). We give some pseudo-umbilical and non pseudo-umbilical examples and we study the link between the proper indexed-biharmonic submanifolds of $\mathbb{S}^n$ and the proper conformal-harmonic submanifolds of $\mathbb{S}^n$.