Classifications of translation and factorable surfaces in the 3-dimensional Galilean space

Study of differential geometry of curves and surfaces in Euclidean, as well as in other non-Euclidean ambient spaces, has a long history. Classical context of the Euclidean space is a source of results which could be transferred to some other geometries. One way of defining new geometries is through Cayley-Klein spaces. They are defined as projective spaces with an absolute figure, a subset of consisting of a sequence of quadrics and planes. Projectivities of the projective space which leave invariant the absolute figure define the subgroup of projectivities called the group of motions of a Cayley-Klein space. By means of the absolute figure, metric relations are also defined and they are invariant under the group of motions. The 3-dimensional Galilean space G_3 is the Cayley-Klein space equipped with the projective metric of signature (0, 0, +, +). The absolute figure of the Galilean geometry consists of an ordered triple (w,f,I), where w is the absolute (ideal) plane, f is the absolute line in w and I is the fixed elliptic involution of points of f. The geometry of Galilean Relativity acts like a ''bridge'' from Euclidean geometry to special Relativity. Galilean space is the space of Galilean Relativity. Galilean space has been largely developed by Röschel. Furhermore, Kamenerovic and Sipus studied about Galilean space. Many works were done about the Galilean and pseudo-Galilean spaces. Galiean and pseudo-Galilean spaces has a crucial role in physics. Galilean space-time are very important in non-relativistic physics. Galilean geometry is a non-Euclidean geometry and associated with Galileo's principle of relativity. Cayley-Klein geometries recently have become interesting again because of their importance for other fields, like soliton theory. In the Galilean space, for given points x = ( x_(1,) x_2 〖,x〗_3) and y=(y_1 〖,y〗_2 〖,y〗_3), the Galilean distance is introduced by the absolute figure, namely d(x,y)={█( |y_1-x_1 |, if 〖 x〗_1≠0 or y_1≠0,@√((y_2-x_2 )^2+(y_3-x_3 )^2 ), if x_1=0 and y_1=0. )┤ The lines and planes are categorized up to the absolute figure. Explicitly, a line is said to be non-isotropic (resp. isotropic) if its intersection with the absolute line I is empty (resp. non-empty). Contrary to this, a plane is said to be isotropic if it does not involve I, otherwise it is said to be Euclidean. In other words, an isotropic plane does not involve any isotropic direction. In the affine model of G_3, the Euclidean planes are determined by the equation x=const. Accordingly, a vector is called isotropic if it is involved in the Euclidean plane x=0. A curve given in parametric form x=x(s)=(x_1 (s)〖,x〗_2 (s)〖,x〗_3 (s)) is said to be non-isotropic (or admissible) if nowhere its tangent vector is isotropic, namely x_1^' (s)=dx/ds≠0. Otherwise the curve x is said to be isotropic. In the Galilean 3-space, the vector x=(x_1,x_2 〖,x〗_3) is isotropic if x_1=0 and non-isotropic otherwise. Hence, for standard coordinates (x_1 〖,x〗_2 〖,x〗_3), the x_1-axis is non-isotropic while the〖 x〗_2-axis and the x_3-axis are isotropic. The x_2 x_3-plane, x_1=0, is Euclidean and the x_1 x_2-plane and x_1 x_3-plane are isotropic. The Galilean scalar product of two vectors x = ( x_(1,) x_2 〖,x〗_3) and y=(y_1 〖,y〗_2 〖,y〗_3) is defined by 〈x,y〉={█(〖 x〗_1 y_1, if x_1≠0 or 〖 y〗_1≠0,@x_2 y_2+x_3 y_3, if x_1=0 and y_1=0 .)┤ The Galilean cross product of two vectors x=(x_1 〖,x〗_2 〖,x〗_3 ) and y=(y_1 〖,y〗_2 〖,y〗_3 ) is defined as x Λ y= |■(0&e_2&e_3@x_1&x_2&x_3@y_1&y_2&y_3 )|= (0,〖 x〗_3 y_1 〖-x〗_1 y_(3 ), x_1 y_2-x_2 y_1. A translation surface in G_3 is locally parametrized by φ:I_1×I_2⊂R^2→G_3, φ(u,v)=α(u)+β(v), where α and β denote translating curves. Under the condition that α and β are planar curves, translating surfaces categorized such a surface up to the absolute figure: Type 1: α is planar non- isotropic curve and β isotropic curve, Type 2: α and β are planar non- isotropic curves. Translation surfaces of the Type 1 in the Galilean space can be locally represented by φ(u,v)=(u,v,f(u)+g(v) ). where α(u)=(u,0,f(u) ) is a non-isotropic curve and β(v)=(0,v,g(v) ) is an isotropic curve. Surface of Type 2, i.e., a surface having both tranlated curves non-isotropic φ(u,v)=(u+v,g(v),f(u) ) where α(u)=(u,0,f(u) ) and β(v)=(v,g(v),0) are a non-isotropic curves. A surface M is a factorable surface if it can be parametrized by φ(u,v)=(u,v,f(u)g(v)). In this thesis, translation and factorable surfaces in a 3-dimensional Galilean space G_3 are studied. This thesis consists of four chapters: In the first chapter, firstly, a brief literature review of Galilean spaces is given. Then, the subject and contents of thesis are given. In the second chapter, translation surfaces with constant Gauss curvature and constant mean curvature in G_3 are studied. Some theorems concerning translation in the Galilean space are given. In the third chapter, first, the definition of coordinate finite type submanifold is given. Then the classification of coordinate finite type translation surfaces in a 3-dimensional Galilean space G_(3 )are studied. In the fourth chapter, the factorable surfaces of first kind in a 3-dimensional Galilean space having non-zero constant mean curvature are examined. Equations of this surface are determined. The following theorem is proved. Theorem: If M is a factorable surface of Type 1 of constant mean curvature H≠0 in 3-dimensional Galilean space, then M is congruent to a surface z= ±1/2H √(1-4H^2 (v+c)^2 ), c=const. Then, it is considered factorable surface of G_(3 ) whose coordinate functions are eigenfunctions of the Laplacian of this surface. It is given a theorem concerning coordinate finite type harmonic factorable surface. The following theorem is proved. Theorem: Let M is a factorable surface of Type 1 in a 3-dimesional Galilean space. If M is harmonic, then it is an open part of a plane.