Transformación conforme: un recorrido desde Euclides hasta Ahlfors

Abstract

Over the course of four chapters, a study is made of conformal mappings from the preservation of generalized circles to the preservation of angles. In the first chapter we try to observe this from the Euclidean geometry in order to motivate the reader about the possibility of observing this kind of properties from a more geometrical point of view. The second chapter consists of notions necessary for the development of the work, a study is made of the analytical functions -useful when defining conformality-, the harmonic functions that provide a widely used theorem on the conservation of right angles, to end with the fact that an analytical function must have a non-zero derivative to preserve the angles between the curves and, therefore, it is a conformal transformation. The last two chapters deal with special conformal mappings. The third chapter delves into the Möbius transformations -fundamental in geometric analysis-; and the fourth chapter examines elementary transformations that preserve conformality.