Abstract

Rollings of reductive homogeneous spaces are investigated. More precisely, for a reductive homogeneous space G/H with reductive decomposition \(\mathfrak {{g}} = \mathfrak {{h}} \oplus \mathfrak {{m}}\) , we consider rollings of \(\mathfrak {{m}}\) over G/H without slip and without twist, where G/H is equipped with an invariant covariant derivative. To this end, an intrinsic point of view is taken, meaning that a rolling is a curve in the configuration space Q which is tangent to a certain distribution. By considering an H-principal fiber bundle \(\overline{\pi }:\overline{Q}\rightarrow Q\) over the configuration space equipped with a suitable principal connection, rollings of \(\mathfrak {{m}}\) over G/H can be expressed in terms of horizontally lifted curves on \(\overline{Q}\) . The total space of \(\overline{\pi }:\overline{Q}\rightarrow Q\) is a product of Lie groups. In particular, for a given control curve, this point of view allows for characterizing rollings of \(\mathfrak {{m}}\) over G/H as solutions of an explicit, time-variant ordinary differential equation (ODE) on \(\overline{Q}\) , the so-called kinematic equation. An explicit solution for the associated initial value problem is obtained for rollings with respect to the canonical invariant covariant derivative of first and second kind if the development curve in G/H is the projection of a one-parameter subgroup in G. Lie groups and Stiefel manifolds are discussed as examples.

中文翻译：

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滚动还原均质空间

摘要

研究了还原均匀空间的滚动。更准确地说，对于具有还原分解的还原齐次空间G / H \(\mathfrak {{g}} = \mathfrak {{h}} \oplus \mathfrak {{m}}\)，我们考虑\(\ mathfrak {{m}}\)在G / H上无滑移且无扭曲，其中G / H配备不变的协变导数。为此，采取了一个内在的观点，即滚动是配置空间Q中与某个分布相切的曲线。通过考虑配备合适主连接的配置空间上的H主纤维束\(\overline{\pi }:\overline{Q}\rightarrow Q\) ，滚动\(\mathfrak {{m}}\ G / H上的)可以用\(\overline{Q}\)上的水平提升曲线来表示。\(\overline{\pi }:\overline{Q}\rightarrow Q\)的总空间是李群的乘积。特别是，对于给定的控制曲线，这种观点允许将G / H上的\(\mathfrak {{m}}\)的滚动表征为\上的显式时变常微分方程 (ODE) 的解。(\overline{Q}\)，即所谓的运动学方程。如果G / H中的展开曲线是G中单参数子群的投影，则针对第一类和第二类规范不变协变导数，可以获得相关初始值问题的显式解。作为例子讨论了李群和斯蒂菲尔流形。