Conexiones sobre Geometría Semi-Riemanniana y Coeficientes de Christoffel

Abstract

This degree paper presents:  The first fundamental form and how from it generate the tensor geometries Riemanniana and Semi-Riemanniana demonstrating certain examples.  A Semi-Riemannian metric on R ^ (n + 1) using some quadratic form and showing that the parallel transport corresponding to the Levi-Civita connection of the metric described coincides with the usual parallel transport of R ^ (n + 1) , This Semi-Remannian metric is called the Lorenz metric for n = 3 which appears naturally in relativity.  The gamma functions which are called the connection coeffi- cients or Christoffel symbols of the connection, calculated on certain differentiable varieties with an associated Semi-Riemannian metric.  The Levi-Civita connection is the only connection where it satisfies that the torsion tensor is zero (symmetric) and that the connection is compatible with the metric known as the fundamental theorem of Semi-Riemannian geometry.  Several examples are considered on the hyperbolic plane such as: 1. Hyperboloid model. 2. Model of the Poincaré disk. 3. Hemispheric model. 4. Model of the upper plane of Poincaré. Where, the Christoffel coefficients, the associated metric, Riemann metric tensor, Ricci curvature tensor and the scalar curvature tensor are calculated by means of the SageMath version 7.6 program.