论文标题
cartan群中的分子问题中的共轭时间
Conjugate time in sub-Riemannian problem on Cartan group
论文作者
论文摘要
Cartan组是等级2和步骤3的自由lie Lie Group。我们考虑在其Lie代数的第一层中定义的cartan群上的左右不变的子伊曼尼亚问题。这个问题给出了与生长矢量$（2,3,5）$的任意次 - 里曼人问题的近似近似。在以前的作品中，我们描述了cartan群上亚里曼尼亚问题的一组对称性，以及相应的麦克斯韦时间 - 对称地球学的第一次相互互相相交。众所周知，在麦克斯韦时间之后，测量学并不是全球最佳的。在这项工作中，我们研究了卡坦组的当地大地测量。我们证明，沿着测量的第一个共轭时间不小于与对称组相对应的麦克斯韦时间。我们表征了第一个共轭时间等于第一个麦克斯韦时间的大地测量学。此外，我们描述了第一个共轭时间在无限值附近的连续性。
The Cartan group is the free nilpotent Lie group of rank 2 and step 3. We consider the left-invariant sub-Riemannian problem on the Cartan group defined by an inner product in the first layer of its Lie algebra. This problem gives a nilpotent approximation of an arbitrary sub-Riemannian problem with the growth vector $(2,3,5)$. In previous works we described a group of symmetries of the sub-Riemannian problem on the Cartan group, and the corresponding Maxwell time -- the first time when symmetric geodesics intersect one another. It is known that geodesics are not globally optimal after the Maxwell time. In this work we study local optimality of geodesics on the Cartan group. We prove that the first conjugate time along a geodesic is not less than the Maxwell time corresponding to the group of symmetries. We characterize geodesics for which the first conjugate time is equal to the first Maxwell time. Moreover, we describe continuity of the first conjugate time near infinite values.
