Fundamentals of Riemannian geometry and its evolution : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University, Palmerston North, New Zealand
In this thesis we study the theory of Riemannian manifolds: these are smooth manifolds equipped with Riemannian metrics, which allow one to measure geometric quantities such as distances and angles. The main objectives are: (i) to introduce some of the main ideas of Riemannian geometry, the geometry of curved spaces. (ii) to present the basic concepts of Riemannian geometry such as Riemannian connections, geodesics, curvature (which describes the most important geometric features of universes) and Jacobi fields (which provide the relationship between geodesics and curvature). (iii) to show how we can generalize the notion of Gaussian curvature for surfaces to the notion of sectional curvature for Riemannian manifolds using the second fundamental form associated with an isometric immersion. Finally we compute the sectional curvatures of our model Riemannian manifolds - Euclidean spaces, spheres and hyperbolic spaces.