Depiction Of Sub-riemannian Geodesic

In this paper we study the geodesies in sub-Riemannian manifolds (M, D,g), where M is a n dirncntional smooth manifold and D is k dimentional smooth horizontal distribution endowed with a smooth positive definite metric tensor y and k n. we discuss two different definitions about such geodesies and the relations between them in this paper. At the same time we classify the sub-Riemannian geodesics into normal geodesics and abnormal geodesics according to whether they are the projcctions of the solutions to the Hamilton equation of sub-Riemannian Hamiltonian or not, classify into regular minimizers and singular minimizers from whether the differential of the endponit maps is surjective at minimizers or not. On the other hand, we get a characteristic for sub-Riemannian geodesies from the view of Optimal Control; we also get the necessary conditions for singular geodesies and the necessary and sufficient conditions for singular curves (i.e. singular points of the endpoint map).