Affine translation surfaces

This thesis has been edited in three chapters. In the first chapter; previous studies about translation surfaces and their background were given. In the second chapter; some fundemental definitions and theorems which often used in Euclidean, Hyperbolic, Galilean and pseudo-Galilean spaces were expressed. In the third chapter; affine translation surfaces with constant curvature in 3-dimensional Euclidean space were investigated and some non-existence theorems for such surfaces were obtained. Parallel surfaces of affine translation surfaces in were investigated. Gaussian and mean curvatures of affine translation surfaces in 3-dimensional Hyperbolic space were obtained and the condition of these surfaces to be minimal were examined. The mean curvatures of affine translation surfaces in 3-sphere were investigated. Affine translation surfaces defined in 4-dimensional Euclidean space and the curvatures were calculated for these surfaces. Also, Gaussian and mean curvatures of affine translation surfaces in 3-dimensional pseudo-Galilean space were calculated. Key Words: Translation surfaces, affine translation surfaces, Euclidean space, Hyperbolic space, Galilean and pseudo-Galilean space.