Topics of Topology and Geometry of Manifolds
Code
11645
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
João Pedro Bizarro Cabral
Weekly hours
1
Total hours
14
Teaching language
Português
Objectives
After completing this course the students will know the fundamentals of Differential Topology and Geometry, in particular, the notion of tangent and cotangent space to a manifold, vector field, differential form, Lie derivative, exterior derivative, integration on manifolds, Stokes'' theorem, vector bundles, and connections and holonomy. It will be emphasized not only that the students know how to prove the main results but also that they know how to calculate in specific examples.
Subject matter
1. Revision of differential calculus, multilinear algebra and general topology.
2. Differential manifolds. Local charts. Differentiable functions. Submanifolds. Whitney''s theorem.
3. Tangent and cotangent space. Vector and tensor fields. Flux of a vector field. Lie bracket. Lie derivatives. Exterior derivatives. Frobenius theorem. Lie groups.
4. Orientation. Integration of differential forms. Stokes theorem.
5. Vector bundles. Connections on Manifolds. Covariant derivative. Parallel transport. Geodesics. Torsion and curvature. Riemannian manifolds. Levi-Civita connection.
6. D''Rham cohomology. Invariance under homotopy. Degree of a map. Relationship between degree and integral. Index of a vector field. Euler characteristic.
7. Principal fibre bundles. Connections and holonomy.
Sections 6 and 7 can be taught alternatively, or partially. Parts of the remaining sections can be omitted, depending on students'' background.
Bibliography
Andrew McInerne, First steps in Differential Geometry, Springer, 2013
Vladimir A. Zorich, Mathematical Analysis II, Springer, 2016
D. Barden, Ch. Thomas, An Introduction to Differential Manifolds, Imperial College Press, 2003.
B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry-Methods and Applications: Part II, the Geometry and Topology of Manifolds, Springer, 1985.
S. Lang, Fundamentals of differential geometry, Springer, 1999.
J.W. Milnor, Topology from the differentiable viewpoint. Princeton University Press, Princeton, NJ, 1997.
M. Perdigão de Carmo, Geometria Riemanniana, IMPA, 1988.
F.W. Warner, Foundations of differentiable manifolds and Lie groups, Springer, 1983.
Teaching method
Tutorial orientation of the student, with indication of which matters to study and give lectures about, problems to solve resolution and clarification of doubts about theory and problems.