On the differential geometry of isometries

This thesis consists of four chapters. The first chapter is the introduction part of thethesis.In the second chapter, fundamental concepts have been given to understand the topicwell.In the third chapter, the rigid motions in Rn+1 have been defined, and rigid motionexamples such as translation, rotation, and reflection have been introduced. Then,theorems and conclusions related to congruent surfaces have been introduced and the invariantsbetween these surfaces have been investigated.In the fourth chapter, the intrinsic geometric properties derived with the help oflength measures of a curve have been given, several examples related to local isometryhave been studied. Moreover, it has been shown that some properties of the surface geometryare isometric invariant whereas others are not. Later the relationships betweenisometries and covariant derivatives have been given. Finally, the relationships betweenRiemann metrics g and isometries have been given. In addition to this, some examplesand theorems related to Riemann metrics have been given.KEYWORDS: Rigid motion, Intrinsic geometry, Isometry, Local isometry.