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Differential Geometry — 1st Semester 2017/2018


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


Lecturer: Slvia 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 first class will be on September 18 at 13h in room P8.

The second class will be on September 20 at 16h in room 4.35 (4th floor) in the Math Building.

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, Poincar'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 (verso em portugus: Lies 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




Links

Other Differential Geometry websites:

1st Semester 2014/2015

1st Semester 2013/2014

1st Semester 2003/2004



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