The Existence And Smoothness Of Sub-Riemannian Geodesics

We investigate geodesics in sub-Riemannian manifold (M, D, g), where M is a smooth manifold of dimension n, g is a positive definite metric on a bracket generating distribution D. It is well known that all geodesics are extremals, and that these extremals are either "normal" or "abnormal". Normal extremals are minimizing, so we called that normal extremals as nor-mal geodesics. The question whether every geodesic was a normal extremal remained open for several years, and was settled by R.Montgomary until1991, who had exhibited a coun-terexample. A geodesic which is not a normal extremal is called a singular geodesic, which depends not on metric and only on the distribution. What are the distributions must exist<not) the singular geodesics? In this paper, we proved that there do no exist any singular geodes ic in a Carnot group G if the type of D is (2,1,…,1) or(2,1,…,1,2). This immediatelys leads to that there are only normal geodesics in the Goursat manifolds. As one corollary, we also obtain that there are only normal geodesics when dimG≤5. Besides, we construct a class of Carnot groups whose distributions satisfy Condition (B1)(see Definition3.2),and on which there exist singular geodesics. We conclude that the dimension n=5is the border line of existence and nonexistence for singular geodesics.Related to the existence of singular geodesics, a natural problem arise:are all sub-Rie-mannian geodesics smooth? Since normal minimizers are automatically smooth, the problem reduces to:are singular geodesics smooth? The problem is an open problem till now. In this paper, we prove that all sub-Riemannian geodesics are smooth if the distribution D sat-isfies [D,[D, D]] c Dand [K,[K, K]] c K for any proper sub-distribution K of D. This re-sult is a generalization of that of Montgomery[31]. According to the result, we obtain some distributions in Lie groups which ensure all geodesics smooth. Further, we prove that all sub-Riemannian geodesics are smooth if the rank-3distribution D satisfies Condition (B2) see Definition4.1). Then, we will construct a Lie group G whose3-dimensional distribution D satisfies condition(B2), and there exist strictly abnormal extremals in G. Finally, we discuss the relationship between singular geodesics and rigid curves. We can see that lots of sing-ular geodesics are rigid curves [29][32]. Whether or not all strictly abnormal minimizers are rigid curves? We prove that not all singular geodesics are rigid curves and we can easily construct a singular geodesic which is not a rigid curve.If we change the sub-Riemannian metric to an indefinite nondegenerate metric on the dis-tribution D, it will produce another different geometry.the simple case is the Lorentzian metric whose the index is1, and it is called a sub-Lorentzian manifold[60,55,52]. There are three classes of geodesics:time-like geodesics, space-like geodesics and null geodesics. The discus-sion of the sub-Lorentzian geodesics is more complicate than sub-Riemannian geodesics. In this paper, we discuss geodesics in the Heisenberg group H" with a Lorentzian metric. We first study the reachable sets by the time-like future directed curves. Second, we give a complete description of the Hamiltonian geodesics. Third, we compute the time-like conjugate locus of the origin.