Geodesic On A Class Of Step Number 2 (k +1) Sub-riemannian Manifold

In this paper we study the geodesics in sub-Riemannian manifold (M,D,g) ,where M(?)R~3=R_x~2×R_t is a three dimentional smooth manifold , D is a two dimentional smooth horizontal distribution generated by vector fieldsinteger, and g is a positive definite metric defined on D .We prove that there exists at least one abnormal geodesic connecting the origin and a distant point . We give out the number of the geodesies that connect the origin and a point which is on t axis and the length of the associated geodesics,at the same time we obtain the shortest geodesic among them.