Extremals in some classes of Carnot groups  

作　　者：HUANG TiRen YANG XiaoPing 

机构地区：[1]Department of Applied Mathematics, Nanjing University of Science ＆ Technology, Nanjing 210094, China

出　　处：《Science China Mathematics》2012年第3期633-646,共14页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No.10771102)

摘　　要：Let G be a Carnot group and D={e 1,e 2 } be a bracket generating left invariant distribution on G.In this paper,we obtain two main results.We first prove that there only exist normal minimizers in G if the type of D is (2,1,...,1) or (2,1,...,1,2).This immediately leads to the fact that there are only normal minimizers in the Goursat manifolds.As one corollary,we also obtain that there are only normal minimizers when dim G 5.We construct a class of Carnot groups such as that of type (2,1,...,1,2,n 0,...,n a) with n 0 1,n i 0,i=1,...,a,in which there exist strictly abnormal extremals.This implies that,for any given manifold of dimension n 6,we can find a class of n-dimensional Carnot groups having strictly abnormal minimizers.We conclude that the dimension n=5 is the border line for the existence and nonexistence of strictly abnormal extremals.Our main technique is based on the equations for the normal and abnormal extremals.Let G be a Carnot group and D={e 1,e 2 } be a bracket generating left invariant distribution on G.In this paper,we obtain two main results.We first prove that there only exist normal minimizers in G if the type of D is （2,1,...,1） or （2,1,...,1,2）.This immediately leads to the fact that there are only normal minimizers in the Goursat manifolds.As one corollary,we also obtain that there are only normal minimizers when dim G 5.We construct a class of Carnot groups such as that of type （2,1,...,1,2,n 0,...,n a） with n 0 1,n i 0,i=1,...,a,in which there exist strictly abnormal extremals.This implies that,for any given manifold of dimension n 6,we can find a class of n-dimensional Carnot groups having strictly abnormal minimizers.We conclude that the dimension n=5 is the border line for the existence and nonexistence of strictly abnormal extremals.Our main technique is based on the equations for the normal and abnormal extremals.

关 键 词：Carnot group EXTREMAL MINIMIZER 

分 类 号：O211.6[理学—概率论与数理统计] O177[理学—数学]