Um breve estudo sobre a geometria diferencial de superfícies em R3

Abstract

The purpose of this work is to make a brief study on the differential geometry of surfaces in R3, with the objective of demonstrating the Gauss-Bonnet theorem in its local and global version. This relevant result relates the geometry and topology of surfaces in R3 and has very interesting consequences. Through it, it is possible to give an answer to an ancient problem of determining whether Euclid’s fifth postulate is an axiom or a theorem. In fact, what is obtained is that there is no harm in denying the fifth postulate, that is, to suppose that there may be more than one or no parallel line to a line r passing through a point p outside of r. What is found are "brave new worlds"that have different geometries from the Euclidean one.