摘要
Let G be a connected Lie group and D be a bracket generating left invariant distribution.In this paper,first,we prove that all sub-Riemannian minimizers are smooth in Lie groups if the distribution D satisfies [D,[D,D]]D and [K,[K,K]]K for any proper sub-distribution K of D.Second,we prove that all sub-Riemannian minimizers are smooth in Lie group if the rank-3 distribution D satisfies Condition（B）.Third,we discuss characterizations of normal extremals,abnormal extremals,rigid curves,and minimizers on product sub-riemannian Lie groups.We prove that not all strictly abnormal minimizers are rigid curves and construct a strictly abnormal minimizer which is not a rigid curve.
Let G be a connected Lie group and D be a bracket generating left invariant distribution.In this paper,first,we prove that all sub-Riemannian minimizers are smooth in Lie groups if the distribution D satisfies [D,[D,D]]D and [K,[K,K]]K for any proper sub-distribution K of D.Second,we prove that all sub-Riemannian minimizers are smooth in Lie group if the rank-3 distribution D satisfies Condition（B）.Third,we discuss characterizations of normal extremals,abnormal extremals,rigid curves,and minimizers on product sub-riemannian Lie groups.We prove that not all strictly abnormal minimizers are rigid curves and construct a strictly abnormal minimizer which is not a rigid curve.
基金
Supported by National Natural Science Foundation of China(Grant Nos.11071119 and 11401531)
