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Differential Geometry — 1st Semester 2018/2019

(for students enrolled in the Master or PhD Mathematics Programs)

Lecturer:	Sílvia Anjos

	Email: sanjos@math.ist.utl.pt

	Office: Mathematics Building, 4th floor, room 4.27

Classes:	Mondays and Tuesdays from 11h to 13h in room P9 (Math Building)
Office Hours:

Announcements

The second exam will take place on January 25, 2019, at 10h, in room P9.

Exam 1 - January 11, 2019

Exam 2 - January 25, 2019

Here are some sample exams: Exam 1 - January 9, 2018, Exam 2 - January 31, 2018

Homework, exam and final grades

Syllabus

Foundations 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.

Bibliography

Recommended Bibliography

R. L. Fernandes, Differential Geometry (versão em português: Lições de Geometria Diferencial)

Other Recommended Bibliography

Warner, 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)

Evaluation

Weekly Homeworks assignments (50% of the final grade) and Final Exam (50% of the final grade).

Homework Assignments

Homework 1, due on Tuesday, September 25

Homework 2, due on Tuesday, October 2

Homework 3, due on Tuesday, October 9

Homework 4, due on Tuesday, October 16

Homework 5, due on Tuesday, October 23

Homework 6, due on Tuesday, October 30

Homework 7, due on Tuesday, November 6

Homework 8, due on Tuesday, November 13

Homework 9, due on Tuesday, November 20

Homework 10, due on Tuesday, November 27

Homework 11, due on Tuesday, December 4

Homework 12, due on Tuesday, December 11

Homework 13, due on Tuesday, December 18

Homework 14 - Connections on Principal Bundles, due on Friday, January 11