The Correspondence Between Focal Surfaces Of Rectilinear Congruence In Minkowski Space E₁～3

The theory of rectilinear congruence is an important field in differential geometry. In chapter 3 of the thesis we study the properties of rectilinear congruence in Minkowski space E₁³ on the basis of [1]. In chapter 4 and 5 we go on investigating the rectilinear congruences T which are generated by the tangent lines to a family of curvature lines or geodesic lines on a space-like surface S in E₁³. We focus on the correspondence of the two focal surfaces and generalize the results of A.Abdel-Baky and B.J.papantonion. The main results obtained in the thesis can be summarized as follows:1. Theorem 3.3 For timelike congruence the focal surfaces are timelike.The focal planes, distribution planes, main planes are also timelike and the two main(distribution) planes are M-orthogonal. The angle between distribution plane and main plane is Ï€/4 or 3Ï€/42. Theorem 3.4 For spacelike congruence if the main surfaces exist then one of the focal surfaces is spacelike the other is timelike, the same as the two facal planes and two main planes. The main planes are orthogonal. If the distribution surfaces exist the two focal surfaces and facal planes are both spacelike or timelike. One of the distribution plane is spcelike the other is timelike. The distribution planes are orhtogonal.3. Theorem 4.6 Let Ψ,(Ψ|ˉ), are two focal surfaces of the timelike normal congruence establishing a one-to-one mapping between the focals on the same normal line of the surface S. K₁,K₂ are the main parameters at the corresponding points; R,R are the radiuses of the base surface S, then:(l)The necessary and sufficient condition that the mapping between Ψ,(Ψ|ˉ) preserves the asymptotic lines is that S is Weingarten surface.(2)The necessary and sufficient condition that the mapping between Ψ,(Ψ|ˉ) preserves the curvature lines is that S is Weingarten surface satisfying R — R = const. Meanwhile K₁ = K₂ =positive constant. (3)If Ψ,(Ψ|ˉ) are developable surfaces then S is isometric to a plane.4. Theorem 5.3 For spacelike congruence T generated by the tangents to the geodesic lines (not straight lines) on a spacelike surface in E₁², the necessary and sufficient condition that the mapping between the (non-degenerate)focal surfaces preserve the asymptotic lines is thatthe Gauss curvature K,K of the two focal surfaces satisfies KK = — g4. q is geodesic curvature radius of the line which is orthogonal to the geodesic line.5. Theorem 5.4 The spacelike congruence T generated by the tangents to the geodesic lines on a spacelike surface is normal congruence. If the surface which is perpedicular to the lines of the congruence is minimal surface then the two focal surfaces preserve the asymptotic lines and the Gauss curvature K,K satisfies KK = —qA. q is geodesic curvature radius of the line which is orthogonal to the geodesic line.