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Differential Geometry — 1st Semester 2016/2017

(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 from 13h to 15h in room Q4.7 and Wednesdays from 14h to 16h in room P1.
Office Hours:	Tuesdays from 16h to 18h

Announcements

The first class will be on September 19 at 13h in room Q4.7 (4th floor of the Chemistry Tower).

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)

Optional 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 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