论文标题
$ \ Mathbb {r}^3 $中兼容的导演字段
Compatible director fields in $\mathbb{R}^3$
论文作者
论文摘要
液晶的组成部分之间的几何形状和相互作用（负责在流体中诱导部分顺序的）可能会在本地偏爱未在$ \ mathbb {r}^3 $中实现的尝试阶段。尽管50年前确定了与$ \ mathbb {r}^3 $几何形状不兼容的状态，但兼容状态的收集仍然鲜为人知，而且表征尚未得到很好的特征。最近，使用移动框架的方法得出了三维导演字段的兼容性条件。这些兼容性条件采用六个差异关系的形式，在本地表征了董事场。在这项工作中，我们使用使用矢量计算的更透明的方法来重新启动这些方程。然后，我们使用这些方程来表征广泛的兼容阶段。
The geometry and interactions between the constituents of a liquid crystal, which are responsible for inducing the partial order in the fluid, may locally favor an attempted phase that could not be realized in $\mathbb{R}^3$. While states that are incompatible with the geometry of $\mathbb{R}^3$ were identified more than 50 years ago, the collection of compatible states remained poorly understood and not well characterized. Recently, the compatibility conditions for three-dimensional director fields were derived using the method of moving frames. These compatibility conditions take the form of six differential relations in five scalar fields locally characterizing the director field. In this work, we rederive these equations using a more transparent approach employing vector calculus. We then use these equations to characterize a wide collection of compatible phases.
