Harmonicity and submanifold maps

The aim of this paper is fourfold. Firstly, we introduce and study the f-ultra-harmonic maps. Secondly, we recall the geometric dynamics generated by a first order normal PDE system and we give original results regarding the geometric dynamics generated by other first order PDE systems. Thirdly, we determine the Gauss PDEs and the fundamental forms associated to integral manifolds of first order PDE systems. Fourthly, we change the Gauss PDEs into a geometric dynamics on the jet bundle of order one, showing that there exist an infinity of Riemannian metrics such that the lift of a submanifold map into the first order jet bundle to be an ultra-potential map.