Um breve estudo sobre o transporte paralelo, geodésicas e a aplicação exponencial

Abstract

Geodesics are curves on a regular surface that have the property of locally minimizing length, that is, if two points are close together, the curve that has the shortest length connecting these two points is a geodesic. They are roughly the "straight lines"of the surface, as they have a constant velocity vector norm, and are zero acceleration curves. We can arrive at these curves through the solution of a variational problem, or following the "path of Geometry"in which we define geodesics as a curve whose field of tangent vectors is parallel. The study of these curves on a surface leads us to the knowledge of several important geometric properties, in addition to the development of new machinery, such as special coordinate systems, for example, which facilitate the study of surfaces and help in the calculation of their important geometric structures. In this work we will make a brief study about parallel transport, Geodesics and the exponential map and its properties. We will study the notion of a covariant derivative, and how we parallel transport vectors along curves. With this idea of parallelism, we will define geodesics as a curve that has a field of parallel tangent vectors, we will study some properties of these curves and the geodesic curvature of curves on surfaces. Finally, we will study the exponential map, the normal coordinate system and the geodesic polar coordinate system and we will use this one to, among other things, show that geodesics have the property of locally minimizing the length.