Bent core (or banana shaped) liquid-crystal-forming-molecules locally favor an ordered state of zero splay and constant bend. Such a state, however, cannot be realized in the plane and the resulting liquid-crystalline phase is frustrated and must exhibit some compromise of these two mutually contradicting local intrinsic tendencies. This constitutes one of the most well-studied examples in which the intrinsic geometry of the constituents of a material gives rise to a geometrically frustrated assembly. Such geometric frustration is not only natural and ubiquitous but also leads to a striking variety of morphologies of ground states and exotic response properties. In this work we establish the necessary and sufficient conditions for two scalar functions, s and b to describe the splay and bend of a director field in the plane. We generalize these compatibility conditions for geometries with non-vanishing constant Gaussian curvature, and provide a reconstruction formula for the director field depending only on the splay and bend fields and their derivatives. Finally, we discuss optimal compromises for simple incompatible cases where the locally preferred values of the splay and bend cannot be simultaneously achieved.

中文翻译：

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二维定向场的几何受挫性和相容性条件

弯曲的核（或香蕉状）液晶形成分子局部倾向于零展开和恒定弯曲的有序状态。然而，这种状态不能在平面上实现，并且所产生的液晶相被挫败，并且必须表现出这两个相互矛盾的局部固有趋势的折衷。这构成了研究最深入的例子之一，其中材料成分的固有几何形状引起了几何上受挫的装配。这种几何上的挫折不仅是自然而普遍的，而且还导致了基态和奇异响应特性的惊人变化。在这项工作中，我们为两个标量函数s和b建立了充要条件描述平面中指向矢场的张开和弯曲。我们针对不具有恒定高斯曲率的几何条件推广了这些相容性条件，并仅根据扩展场和弯曲场及其导数提供了指向矢场的重构公式。最后，我们讨论了无法同时实现张开和弯曲的局部首选值的简单不兼容情况的最佳折衷方案。