一类步数为2(k+1)的次黎曼流形上测地线的研究  

A Study of Geodesics on a Certain Step 2(k+1) Sub-Riemannian Manifold

作　　者：张学华[1] 杨孝平[2] 

机构地区：[1]黄山学院数学系,安徽黄山245041 [2]南京理工大学理学院,江苏南京210094

出　　处：《大学数学》2010年第2期80-86,共7页College Mathematics

基　　金：安徽省高等学校省级优秀青年人才基金项目(2010SQRL169);黄山学院科研发展基金资助(2006xkjq002)

摘　　要：构造了一类步数为2(k+1)的次黎曼流形,给出其上连接原点和t轴上一点测地线的条数和相应测地线的长度,同时得到其中最短的测地线.We study the geodesics in sub-Riemannian manifold （M, D, g）, where M≌R^3= Rx^ × R, 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. We give out the number of the geodesics 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.

关 键 词：次黎曼流形 水平分布 正规测地线 奇异测地线 

分 类 号：O186.16[理学—数学]