一类次黎曼流形的步数计算

The Calculation on the Step of a Certain Sub-Riemannian Manifold

构造一类次黎曼流形(M,D,g)并计算出此类次黎曼流形的步数,这里M≡R^3=R_x^2×R_t是3维光滑流形,D是由切向量场Y_1,Y_2生成的2维光滑水平分布,其中Y_1=1/((1+|x|^(4k+2))^(1/2))(/(x_1)+2x_2|X|^(2k)/(t)),Y_2=1/((1+|x|^(4k+2))^(1/2)) (/(x_2)-2x_1|X|^(2k)/(t)),k≥0是整数,g是定义在D上的正定度量。

In this paper we give a certain sub-Riemannian manifold （M,D,g） and the step of it, where M≡R^3=Rx^2×Rt is a three dimentional smooth manifold, D is a two dimentional smooth horizontal distribution generated by vector fields Y1=1/√1＋｜x｜^4k＋2（ /x1＋2x2｜x｜^2k /t）,Y2=1/√1＋｜x｜^4k＋2 （ / x2-2x1｜x｜^2k / t）,k≥0 is an integer, and g is a positive definite metric defined on D.