内禀横摇存在的几何条件
Geometric conditions for the existence of an intrinsic rolling
摘要：
对于两个不同的黎曼流形中的两条曲线何时可以看作是一个流形在另一个流形上滚动而没有扭曲或滑动的轨迹的问题，我们给出了一个完整的答案。我们表明，技术的假设，滚动沿这些曲线存在当且仅当每个曲线的测地曲率重合。利用曲线的反展开式，我们可以将其看作是测地曲率的推广，从而将结果推广到任意绝对连续的曲线.对于在自身上滚动的常截面曲率的流形，两条这样的曲线只能相差等距线。在表面的情况下，我们给出的条件时，循环在流形上的电梯循环的配置空间的滚动。
英文摘要：
We give a complete answer to the question of when two curves in two different Riemannian manifolds can be seen as trajectories of rolling one manifold on the other without twisting or slipping. We show that up to technical hypotheses, a rolling along these curves exists if and only if the geodesic curvatures of each curve coincide. By using the anti-developments of the curves, which we claim can seen of as a generalization of the geodesic curvatures, we are able to extend the result to arbitrary absolutely continuous curves. For a manifold of constant sectional curvature rolling on itself, two such curves can only differ by an isometry. In the case of surfaces, we give conditions for when loops in the manifolds lift to loops in the configuration space of the rolling.
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