Differential Geometry — 1st Semester 2016/2017(for students enrolled in the Master or PhD Mathematics Programs)
AnnouncementsThe first class will be on September 19 at 13h in room Q4.7 (4th floor of the Chemistry Tower). 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) Optional 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 Wednesday, September 28 Homework 2, due on Thursday, October 6 Homework 3, due on Wednesday, October 12 Homework 4, due on Wednesday, October 19 Homework 5, due on Wednesday, October 26 Homework 6, due on Wednesday, November 2 Homework 7, due on Wednesday, November 9 Homework 8, due on Wednesday, November 16 Homework 9, due on Wednesday, November 23 Homework 10, due on Wednesday, November 30 Homework 11, due on Wednesday, December 7 Homework 12, due on Wednesday, December 14 Homework 13, due on Wednesday, January 4 Extra Homework, not mandatory LinksOther Differential Geometry websites: |