scholarly journals Totally geodesic submanifolds of naturally reductive homogeneous spaces

1996 ◽  
Vol 20 (1) ◽  
pp. 181-190 ◽  
Author(s):  
Koji Tojo
Keyword(s):  
Homogeneous Spaces ◽  
Totally Geodesic ◽  
Geodesic Submanifolds ◽  
Totally Geodesic Submanifolds ◽  
Reductive Homogeneous Spaces ◽  
Naturally Reductive

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.

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