卡诺群中测地线的爆破与爆破
Blowups and blowdowns of geodesics in Carnot groups
摘要：
本文给出了测地线的部分正则性结果（即，等距图像的间隔）在任意的子黎曼流形和子Finsler流形。我们的战略是研究无穷小和渐近性质的测地线在卡诺集团配备了任意子芬斯勒度量。证明了在幂零阶较低的群中卡诺测地线的切线是测地线。也就是说，每个卡诺群中的每个测地线的每个爆破曲线仍然是群中模其最后一层的测地线。作为推论，我们得到在每个次黎曼流形中，任何测地线的任意$s$次迭代切线都是一条直线，其中$s$是所讨论的次黎曼流形的步长.用类似的方法，我们还证明了次黎曼卡诺群中测地线的吹倒曲线包含在低秩子群中。后者的结果也扩展到粗糙测地线。
英文摘要：
This paper provides some partial regularity results for geodesics (i.e., isometric images of intervals) in arbitrary sub-Riemannian and sub-Finsler manifolds. Our strategy is to study infinitesimal and asymptotic properties of geodesics in Carnot groups equipped with arbitrary sub-Finsler metrics. We show that tangents of Carnot geodesics are geodesics in some groups of lower nilpotency step. Namely, every blowup curve of every geodesic in every Carnot group is still a geodesic in the group modulo its last layer. Then as a consequence we get that in every sub-Riemannian manifold any $s$ times iterated tangent of any geodesic is a line, where $s$ is the step of the sub-Riemannian manifold in question. With a similar approach, we also show that blowdown curves of geodesics in sub-Riemannian Carnot groups are contained in subgroups of lower rank. This latter result is also extended to rough geodesics.
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