Intrinsic Geometry and Director Reconstruction for Three-Dimensional Liquid Crystals
TL;DR: In this paper, the intrinsic geometry of elastic distortions in three-dimensional nematic liquid crystals is described and necessary and sufficient conditions for a set of functions to represent these distortions by describing how they couple to the curvature tensor.
Abstract: We give a description of the intrinsic geometry of elastic distortions in three-dimensional nematic liquid crystals and establish necessary and sufficient conditions for a set of functions to represent these distortions by describing how they couple to the curvature tensor. We demonstrate that, in contrast to the situation in two dimensions, the first-order gradients of the director alone are not sufficient for full reconstruction of the director field from its intrinsic geometry: it is necessary to provide additional information about the second-order director gradients. We describe several different methods by which the director field may be reconstructed from its intrinsic geometry. Finally, we discuss the coupling between individual distortions and curvature from the perspective of Lie algebras and groups and describe homogeneous spaces on which pure modes of distortion can be realised.
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TL;DR: In this paper, the author outlines what is known to the author about the Riemannian geometry of a Lie group which has been provided with a metric invariant under left translation.
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01 Jan 2008TL;DR: A comprehensive introduction to contact topology is given in this article, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds.
Abstract: This text on contact topology is a comprehensive introduction to the subject, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds. Starting with the basic differential topology of contact manifolds, all aspects of 3-dimensional contact manifolds are treated in this book. One notable feature is a detailed exposition of Eliashberg's classification of overtwisted contact structures. Later chapters also deal with higher-dimensional contact topology. Here the focus is on contact surgery, but other constructions of contact manifolds are described, such as open books or fibre connected sums. This book serves both as a self-contained introduction to the subject for advanced graduate students and as a reference for researchers.
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TL;DR: In this paper, all invariant functions of the group generators (generalized Casimir operators) are found for real algebras of dimension up to five and for all real nil-potent algebras of dimension six.
Abstract: All invariant functions of the group generators (generalized Casimir operators) are found for all real algebras of dimension up to five and for all real nilpotent algebras of dimension six.
492 citations
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01 Jan 1945
294 citations