On distance preserving mappings

This thesis consist of five chapters. The first chapter is devoted to the introduction section. In the second chapter, some required preparatory notions recalled. In the third chapter, Mazur-Ulam theorem and proof is given. In the fourth chapter, the trabsformations that maintain the unit distance are discussed. In the fifth chapter, the definition of Möbius transformation and basic definitions in hyperbolic geometry are given. Then Beckman -Quarles theorem is proved that hyperbolic geometry.