The Horizontal Harmonic Mapping From The Heisenberg Group To Contact Manifolds

The main purpose of this thesis is to generalize the notion of harmonic maps between Riemannian manifolds to sub-Riemannian manifolds. In the introduction, we give some research background of the thesis and current status related to sub-Riemannian geometry. In chapter two, we review the concept of harmonic maps which are annihilating the trace of the second fundamental form. In chapter three, we review some essential concepts and examples of sub-Riemannian manifold, and recollect the notion of horizontal connection and its properties. In chapter four, after introducing the notion of horizontal induced connection, we give the concept of the horizontal second fundamental form of contact maps. Then we give the notion of horizontal harmonic maps between sub-Riemannian manifolds. In chapter five, we give sufficient and necessary condition satisfied by the horizontal harmonic maps between Heisenberg groups, and prove that the component functions on Heisenberg groups are horizontal harmonic.Compared with harmonic maps on Riemannian manifold where the harmonic maps are harmonic functions when target space and source space are Euclidean space, the results of this thesis are natural.