Differential Geometry — 1st Semester 2017/2018(for students enrolled in the Master or PhD Mathematics Programs)
AnnouncementsThe second exam is on January 31, at 11h30, in room P1. The first exam is on January 9, at 15h, in room P1. Homework, exam and final grades (updated on January 15 to include the grades of the first exam). From the second week classes will be on Mondays and Tuesdays from 13h to 15h in rooms P8 and 4.35 (Math Building) , respectively. 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 Tuesday, September 26 Homework 2, due on Tuesday, October 3 Homework 3, due on Tuesday, October 10 Homework 4, due on Tuesday, October 17 Homework 5, due on Tuesday, October 24 Homework 6, due on Tuesday, October 31 Homework 7, due on Tuesday, November 7 Homework 8, due on Tuesday, November 14 Homework 9, due on Tuesday, November 21 Homework 10, due on Tuesday, November 28 Homework 11, due on Tuesday, December 5 Homework 12, due on Tuesday, December 12 Homework 13, due on Tuesday, January 9 Extra Homework, not mandatory LinksOther Differential Geometry websites: |