Contribution to the geometry of polyharmonic maps

Abstract

Polyharmonic maps of order k are a natural generalization of harmonic maps, for k = 2, this maps are called biharmonic maps. In this thesis we will study the biharmonicity of a vector eld X on a pseudo-Riemannian manifold (M; g) viewed as a map X : (M; g) ! (TM; gS) where gS is the Sasaki metric. More precisely, we establish the formula of the bitension eld of X and we show characterization theorem for X to be biharmonic map, and we describe the relationship between vector elds X that are critical points of the bienergy functional E2 restricted to variations through vector elds, equivalently X are biharmonic vector elds, and vector elds which are biharmonic maps. Moreover, several applications are included.