A Kind of Rectilinear Congruences in the Minkowski 3-Space
Received:May 28, 2007  Revised:March 08, 2008
Key Words: the rectilinear congruence   the focal surfaces   the Minkowski space   the space-like surface   maximal surfaces.  
Fund Project:Specialized Research Fund for the Doctoral Program of Higher Education (No.20050141011).
Author NameAffiliation
HOU Zhong Hua Department of Applied Mathematics, Dalian University of Technology, Liaoning 116024, China 
LI Li College of Stat.& Applied Mathematics, Anhui University of Fin.& Econ., Anhui 233041, China 
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Abstract:
      We consider the rectilinear congruence $T$ generated by the tangents to a one parameter family of geodesics on a space-like surface $S_1$ in the Minkowski $3$-space $E_1^3$, having $S_1$ as one of its focal surfaces. We prove that the two families of torsal surfaces of $T$ touch the second focal surface $S_2$ along the net of orthogonal parametric curves if and only if $S_1$ is developable. We also obtain the necessary and sufficient condition for the correspondence between the points of $S_1$ and $S_2$ at the same rays preserving the net of asymptotic curves. At last, we investigate the orthogonal surface $S$ of $T$. We proved that the correspondence between $S_1$ and $S_2$ preserves the net of asymptotic curves if $S$ is maximal in $E_1^3$.
Citation:
DOI:10.3770/j.issn:1000-341X.2008.04.022
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