Topics in Differential and Riemannian Geometry

Abstract

This project is based on the knowledge of Analysis and Algebraic Topology obtained during the project "Introduction to Algebraic Topology", carried out during the last three semesters, using these concepts to introduce a study of Differential and Riemannian Geometry, concluding the student's undergraduate formation and giving necessary techniques to further graduate studies.The main objectives are not only to consider the fundamentals of these geometries, but also to develop a first study of Exterior Algebra and Differentiable Manifolds, including topics as multilinear maps, exterior forms, tensors, differential forms and Lie groups. Prepared with this preliminary work, it will be possible to approach essential topics of Riemannian Geometry and Grassmann's Algebra, studying problems related to geodesics, curvature tensors, Ricci curvature, Lie group actions in homogeneous spaces, symplectic spaces and the Lagrangian-Grassmannian.In conclusion, the student will deal with the Maslov Index, a symplectic invariant associated with solutions of hamiltonian systems. This index is calculated by counting algebraically the zeroes of the linearized system, and relates to Riemannian Geometry corrisponding to the geometric index of a geodesic, i.e., the number of conjugated points in this curve.