论文标题
无限的螺旋螺旋运动，具有空间形式的定向地球学的固定音调
Infinitesimally helicoidal motions with fixed pitch of oriented geodesics of a space form
论文作者
论文摘要
令L为r^3的所有（未参数）方向线的歧管。我们研究了由L中的曲线在每个瞬间描述的条件下，在无限水平上描述的条件下，控制系统的可控性。实际上，我们通过控制系统在曲率kappa的三维空间形式的所有定向的完整的大地测量上，构成了类似的更通用的问题：kappa = 0，kappa = 0，s^3 for kappa = 1 = 1和kappa = -1的超纤维3空间。当且仅当Alpha ^2不等于Kappa时，我们才能获得系统可控。在带有alpha =（+/-）1的球形情况下，可允许的曲线保留在S^3的固定HOPF纤维纤维中。 在这种情况下，我们还解决并解决了一种肯德尔（又名牛津）问题：找到分段连续曲线的最小开关数量加入了两条任意方向的线，在一些杰出的可允许曲线家族中，有零件。
Let L be the manifold of all (unparametrized) oriented lines of R^3. We study the controllability of the control system in L given by the condition that a curve in L describes at each instant, at the infinitesimal level, an helicoid with prescribed angular speed alpha. Actually, we pose the analogous more general problem by means of a control system on the manifold G_kappa of all the oriented complete geodesics of the three dimensional space form of curvature kappa: R^3 for kappa = 0, S^3 for kappa = 1 and hyperbolic 3-space for kappa = -1. We obtain that the system is controllable if and only if alpha ^2 not equal kappa. In the spherical case with alpha = (+/-) 1, an admissible curve remains in the set of fibers of a fixed Hopf fibration of S^3. We also address and solve a sort of Kendall's (aka Oxford) problem in this setting: Finding the minimum number of switches of piecewise continuous curves joininig two arbitrary oriented lines, with pieces in some distinguished families of admissible curves.
