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
  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 $\mathbb{S}^n$. 
Vincent B\'erard
math.DG