scholarly journals THE GEODESIC RULE FOR HIGHER CODIMENSIONAL GLOBAL DEFECTS

2008 ◽  
Vol 23 (25) ◽  
pp. 2053-2066
Author(s):  
ANTHONY J. CREACO ◽  
NIKOS KALOGEROPOULOS
Keyword(s):  
Liquid Crystals ◽  
Diffusion Equation ◽  
Convex Hull ◽  
Geometric Structure ◽  
Order Parameter ◽  
The Other ◽  
Equation Approach ◽  
Totally Geodesic ◽  
Geodesic Submanifolds ◽  
Totally Geodesic Submanifolds

We generalize the geodesic rule to the case of formation of higher codimensional global defects. Relying on energetic arguments, we argue that, for such defects, the geometric structures of interest are the totally geodesic submanifolds. On the other hand, stochastic arguments lead to a diffusion equation approach, from which the geodesic rule is deduced. It turns out that the most appropriate geometric structure that one should consider is the convex hull of the values of the order parameter on the causal volumes whose collision gives rise to the defect. We explain why these two approaches lead to similar results when calculating the density of global defects by using a theorem of Cheeger and Gromoll. We present a computation of the probability of formation of strings/vortices in the case of a system, such as nematic liquid crystals, whose vacuum is ℝP2.

A continuum theory of disorder in nematic liquid crystals. - II. Intermolecular correlations

1977 ◽  
Vol 353 (1673) ◽  
pp. 261-275 ◽  
Keyword(s):  
Liquid Crystals ◽  
Order Parameter ◽  
Continuum Theory ◽  
The Other ◽  
Wave Number ◽  
Interesting Problem ◽  
Molecular Models ◽  
Partial Agreement ◽  
The Mean ◽  
The Continuum

The continuum theory of nematics suggested in paper I is used to derive an expression for the quantity < P 2 (cos γ ( R ))>, where γ is the angle between the director at two points separated by a distance R . The result tends to the Maier-Saupe limit (S 2 2 , corresponding to no correlations of orientation) for large R , but to unity for small R , while for the value of R corresponding to the mean intermolecular spacing it is about S α 0 2 , with α 0 close to unity. It is suggested that continuum theory may be used to estimate <sin γ ( R )> as well. Two simple molecular models for nematics are discussed in the light of these results, one of them a simplified version of the model on which Maier & Saupe originally based their theory, and the other a steric model of the sort proposed by Onsager. Predictions based upon these models concerning the Frank stiffness constants - in particular, concerning their dependence on the order parameter S 2 at constant volume and temperature and upon wave number q - are found to be in partial but only partial agreement with experiment. An interesting problem concerning the entropy of misalignment and its effect upon the stiffness of a nematic is left unresolved.

scholarly journals A Note on Triple Systems and Totally Geodesic Submanifolds in a Homogeneous Space

1968 ◽  
Vol 32 ◽  
pp. 5-20 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Symmetric Space ◽  
Jordan Algebras ◽  
Riemannian Symmetric Space ◽  
Lie Triple System ◽  
Totally Geodesic ◽  
Triple Systems ◽  
Geodesic Submanifolds ◽  
Totally Geodesic Submanifolds ◽  
Nonassociative Algebras ◽  
Torsion Free

In the study of nonassociative algebras various “triple systems” frequently arise from the associator function and other multilinear objects. In particular Lie triple systems arise in the study of Jordan algebras and a generalization of a Lie triple system arises in Malcev algebras. Lie triple systems also are used to study totally geodesic submanifolds of a Riemannian symmetric space. We shall show how a generalization of Lie triple systems also arises from the study of curvature and geodesies of a torsion free connexion on a manifold and bring out the relation of this to various nonassociative algebras.

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