三维指向矢场的活动标架和相容条件
Moving frames and compatibility conditions for three-dimensional director fields
摘要：
的几何形状和拓扑结构的区域中，一个导演字段是嵌入施加限制的种类支持的取向顺序。这些限制表现为相容性条件，其将描述指向矢场的量与嵌入空间的几何形状相关联。例如，在二维（2D）中，展曲场和弯曲场足以唯一地确定指向矢（直到刚性运动），并且必须符合嵌入流形的高斯曲率中的一个线性关系。在3D中，存在描述指向矢的附加局部场，即驻留在材料内的局部观察者可用的场，以及产生几何挫折的许多不同方式。到目前为止，还不知道需要多少这样的局部场来唯一地描述3D指向矢场，也不知道它们必须满足什么样的兼容关系。在这项工作中，我们直接解决这些问题。我们采用移动标架的方法证明了一个指向矢场完全由五个局部场决定。这些领域被证明是相互关联的，并通过六个微分关系的嵌入空间的曲率。作为我们方法的一个应用，我们描述了所有均匀畸变指向矢场，即，在常曲率流形中，所有局部特征场在空间中是常数的导演。这种阶段的分类最近已经提供了董事在欧几里得空间，其中的纹理对应于叶理的空间由平行全等螺旋。对于非零曲率，我们表明，纯扭曲相位是唯一的解决方案，在正弯曲的空间，而在双曲空间均匀畸变场对应于（不一定平行）全等螺旋线的空间的叶理。进一步的分析，预计也允许构建新的非均匀的导演字段。
英文摘要：
The geometry and topology of the region in which a director field is embedded impose limitations on the kind of supported orientational order. These limitations manifest as compatibility conditions that relate the quantities describing the director field to the geometry of the embedding space. For example, in two dimensions (2D) the splay and bend fields suffice to determine a director uniquely (up to rigid motions) and must comply with one relation linear in the Gaussian curvature of the embedding manifold. In 3D there are additional local fields describing the director, i.e. fields available to a local observer residing within the material, and a number of distinct ways to yield geometric frustration. So far it was unknown how many such local fields are required to uniquely describe a 3D director field, nor what are the compatibility relations they must satisfy. In this work, we address these questions directly. We employ the method of moving frames to show that a director field is fully determined by five local fields. These fields are shown to be related to each other and to the curvature of the embedding space through six differential relations. As an application of our method, we characterize all uniform distortion director fields, i.e., directors for which all the local characterizing fields are constant in space, in manifolds of constant curvature. The classification of such phases has been recently provided for directors in Euclidean space, where the textures correspond to foliations of space by parallel congruent helices. For non-vanishing curvature, we show that the pure twist phase is the only solution in positively curved space, while in the hyperbolic space uniform distortion fields correspond to foliations of space by (non-necessarily parallel) congruent helices. Further analysis is expected to allow to also construct of new non-uniform director fields.
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