Mathematics: Differential Geometry

<p>The course gives an introduction to classic differential geometry, important for further studies in the subject and in relevant areas of physics.Differential geometry is necessary to understand Riemannian geometry, which is an important component in Einstein's general theory of relativity.</p><p>The course treats the geometry of curves and surfaces, especially in three dimensions. In particular, we study the concepts of curvature and torsion. The course covers:</p> <ul> <li>The geometry of curves in Euclidean space, their curvature and torsion and how these determine the curves.</li> <li>The geometry of surfaces in Euclidean space, their first and second fundamental forms, the Gauss map, principal curvatures, Gaussian curvature and mean curvature.</li> <li>Theorema Egregium and a deep analysis of geodesics and their behaviour both locally and globally.</li> <li>Gauss-Bonnet's Theorem: two different local versions and the famous global version.</li> </ul>