Superfícies em RxH²

Abstract

The main objective of this work is to investigate the geometry of surfaces immersed in R×H2 , a non-Euclidean space, that is, with a geometry distinct from that defined in R 3 . To enable this investigation, the first chapter was dedicated to the development of fundamental concepts of classical differential geometry in R 3 , including the definitions of regular and induced metric surfaces, as well as notions of geodesics and surface completeness. Geometric entities such as mean and Gaussian curvatures were also defined. We conclude this chapter with the definition of the pseudosphere, a surface of constant negative Gaussian curvature, which served as a bridge to the exploration of other non-Euclidean spaces. Later, in the second chapter, we introduce the Lorentz-Minkowski space and, within this context, we define the hyperbolic space H2 . We also present some fundamental results about this space, highlighting its completeness in this new space. Then, in the third chapter, we address essential concepts of Riemannian geometry, which generalizes the results of the first chapter to larger spaces. In this context, we explore notions such as Riemannian metric, affine connections and curvatures, with emphasis on the Levi-Civita theorem, a fundamental result for Riemannian geometry. Finally, in the last chapter of this text, we construct the space R × H2 and explore several geometric properties associated with it. In particular, we present results on curvatures in this context, highlighting the famous Gauss theorem, which relates extrinsic curvatures to an intrinsic entity of surfaces in an arbitrary space M3 . Furthermore, we apply these results to concrete examples, illustrating their relevance and applicability.