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Differential Geometry — 1st Semester 2017/2018

(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 13h to 15h in rooms P8 and 4.35, respectively (Math Building)
Office Hours:	Mondays from 15h to 17h

Announcements

The 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.

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 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