A study of intrinsic geometry of surface

This master's thesis consisting of five chapters, The first chapter consists of the introduction and focuses on the situations related to which studies performed on a surface describe the internal properties of the surface. The second part is devoted to basic definitions and concepts that require a better understanding of the subject. The third chapter contains the Lie Operator and Lie algebra structure, which are important concepts of differential geometry. Additionally, the concept of parallel transport between surfaces is also examined here. In the fourth chapter, information about Surface at R3, isometric surfaces, directional derivative regarding the internal approximation of the surface, covariant derivative and basic forms of the surface are presented. In the fifth chapter, information is given about the Gaussian curvature Revisited, the Egregium Theorem, which means "extraordinary theorem" in Latin, and the Gauss Bonnet Theorem.