Some Properties Of Rectilinear Congruence In Minkowski 3-space

The theory of rectilinear congruence is an important field in differential geometry. In this thesis, we deal with the theory of rectilinear congruence in the Minkowski 3-space, and we define the fundamental forms and fundamental elements of the congruences.At first, we classify the rectilinear congruence into spacelike congruence and timelike congruence based on the type of the ray. especially we discuss some properties of timelike congruence. We generalize a theorem about the distribution parameter which have been proved by Su Buchin in 1927.The main result obtained in this thesis about the distribution parameter can be summarized as follows:1. Theorem 3.7 In the tmeklike congruence, let d1u : d1v , d2u : d2v be the orientation of the two main surfaces drawed from the ray l(u, v), 6 be the fixed angle between the third orientation du : dv and d1U : d\v, q be the distance between the middle point and the center point of the ray l(u, v), p be the distribution parameter. Then the p, q and H of the developable surface should be satisfied:Secondly we discuss the normal congruence in M13, where the normal surface 5 is a timelike surfaces. We consider what forms S should have if the mapping between the focal surfaces of the normal congruences of S, preserve the curvature lines. The theorem can be summarized as follows:2. Theorem 4.1 In M13 ,if the mapping between the focal surfaces of the normal congruences of 5, preserve the curvature lines, then(a) 5 is a spherical surface.(b) 5 is isometric to a plane.(c) S is isometric to a surface of revolution.(d) The first fundamental form of S should satisfy the following equation ds2 = (H2 + K)(du2 + dv2), where K, H are the Gauss curvature and the mean curvature of S.