scholarly journals The Geometry of the Space of Oriented Geodesics of Hyperbolic $3$-Space

2009 ◽  
Vol 0064 ◽  
pp. 15
Author(s):  
Nikos Georgiou
Keyword(s):  
Space Of Oriented Geodesics

scholarly journals On area stationary surfaces in the space of oriented geodesics of hyperbolic 3-space

2012 ◽  
Vol 111 (2) ◽  
pp. 187
Author(s):  
Nikos Georgiou
Keyword(s):  
Parameter Family ◽  
Holomorphic Curve ◽  
Kähler Structure ◽  
Stationary Surface ◽  
The Family ◽  
Rotationally Symmetric ◽  
Parallel Surfaces ◽  
Space Of Oriented Geodesics ◽  
Two Parameter

We study area-stationary surfaces in the space $\mathbf{L}(\mathbf{H}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral Kähler structure. We prove that every holomorphic curve in $\mathbf{L}(\mathbf{H}^3)$ is an area-stationary surface. We then classify Lagrangian area-stationary surfaces $\Sigma$ in $\mathbf{L}(\mathbf{H}^3)$ and prove that the family of parallel surfaces in $\mathbf{H}^3$ orthogonal to the geodesics $\gamma\in \Sigma$ form a family of equidistant tubes around a geodesic. Finally we find an example of a two parameter family of rotationally symmetric area-stationary surfaces that are neither Lagrangian nor holomorphic.

scholarly journals On the Space of Oriented Geodesics of Hyperbolic 3-Space

2010 ◽  
Vol 40 (4) ◽  
pp. 1183-1219 ◽  
Author(s):  
Nikos Georgiou ◽  
Brendan Guilfoyle
Keyword(s):  
Space Of Oriented Geodesics
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