Chaturvedi N, Kamien Randall D
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, USA.
Proc Math Phys Eng Sci. 2020 Feb;476(2234):20190824. doi: 10.1098/rspa.2019.0824. Epub 2020 Feb 19.
The Hopf fibration has inspired any number of geometric structures in physical systems, in particular, in chiral liquid crystalline materials. Because the Hopf fibration lives on the three sphere, , some method of projection or distortion must be employed to realize textures in flat space. Here, we explore the geodesic-preserving gnomonic projection of the Hopf fibration, and show that this could be the basis for a new liquid crystalline texture with only splay and twist. We outline the structure and show that it is defined by the tangent vectors along the straight line rulings on a series of hyperboloids. The phase is defined by a lack of bend deformations in the texture, and is reminiscent of the splay-bend and twist-bend nematic phases. We show that domains of this phase may be stabilized through anchoring and saddle-splay.
霍普夫纤维化启发了物理系统中众多的几何结构，特别是在手性液晶材料中。由于霍普夫纤维化存在于三维球面上， ，因此必须采用某种投影或变形方法才能在平面空间中实现纹理。在这里，我们探索了霍普夫纤维化的保测地线心射投影，并表明这可能是一种仅具有展曲和扭曲的新型液晶纹理的基础。我们概述了其结构，并表明它由一系列双曲面上沿直线母线的切向量定义。该相由纹理中缺乏弯曲变形来定义，让人联想到展曲-弯曲和扭曲-弯曲向列相。我们表明，通过锚定和鞍形展曲可以稳定该相的畴。
