Differential Geometry — 1st Semester 2018/2019(for students enrolled in the Master or PhD Mathematics Programs)
AnnouncementsThe classes are on Mondays and Tuesdays from 11h to 13h in room P9 (Math Building) and the first day of classes is September 17. SyllabusFoundations of Differential Manifolds: Manifolds, partitions of unity, tangent space. Submersions, imersions, submanifolds, Whitney Theorem. Foliations. Lie Theory: Vector fields, Lie brackets, Lie derivative. Distributions and Frobenius Theorem. Lie groups, Lie algebras, actions. Differential Forms: Tensor and exterior algebras, differential forms. Cartan's formula, de Rham cohomology, Poincaré's lemma. Orientation, integration over manifolds, homotopy. Stokes Theorem, Mayer-Vietoris sequence. Fiber Bundles: Vector bundles, connections, curvature, metrics. Parallel transport, Riemannian manifolds, geodesics. Characteristic classes, Chern-Weil theory. Gauss-Bonnet Theorem. Principal bundles and Ehresmann connections. BibliographyRecommended BibliographyR. L. Fernandes, Differential Geometry (versão em português: Lições de Geometria Diferencial) Other Recommended BibliographyWarner, Foundations of Differentiable Manifolds and Lie Groups, Springer (1983) Bott and Tu, Differential Forms in Algebraic Topology, Springer (1986) Kobayashi and K. Nomizu, Foundations of Differential Geometry (2 vols.), John Wiley & Sons (1996) EvaluationWeekly Homeworks assignments (50% of the final grade) and Final Exam (50% of the final grade). Homework AssignmentsHomework 1, due on Tuesday, September 25 LinksOther Differential Geometry websites: |