Geodesic. Carnot, Group

In this paper, the geodesics in a class of sub-Riemannian manifolds-Carnot groups are studied. Firstly, we summarize the definitions of sub-Riemannian geodesics which are given from different views, and compare with the different definitions. Secondly, we discuss the definitions of geodesies in Riemannian manifolds, and prove the equivalence of the minimizing geodesies and the straightest geodesies in Riemannian manifold. Then we discuss a class of the simplest Carnot groups-the Heisenberg group, in which the non-equivalence of the minimizing geodesies and the straightest geodesies is established. Finally, we investigate the geodesies in the Engel manifold, and give equations of the minimizing geodesies in such manifold. Based on the equations, the non-equivalence of the minimizing geodesies and the straightest geodesies in Engel manifold is proved.