Differentiable Manifolds

Recommended Prerequisites

Multivariable calculus, elements of general topology.

Teaching Methods

Lectures have an expository character, being up to the professor the choice of the most appropriate way to do it and the degree of participation of the students. As an integral part of the learning process, it may be recommended or required  the solution of exercises, course projects or oral presentations.

Learning Outcomes

The goal of this course is to provide a general knowledge of Differentiable Manifolds  at an advanced level. It is intended that the students become familiar with the main basic techniques and results of this area of Mathematics, or at least attain enough familiarity with some of  them to be able to acquire others by themselves that may later prove to be useful.

Work Placement(s)

No

Syllabus

Variedades diferenciáveis: estrutura local, subvariedades, Teorema de Sard, transversalidade, campos de vetores e fluxos. Noção de fibrado, fibrados tangente e cotangente de uma variedade. Derivada de Lie de campos de vectores. Álgebras de Lie. Grupos de Lie (clássicos). Espaços homogéneos. Formas diferenciais, derivada exterior. Formas simpléticas. Integração em variedades.  Teorema de Stokes.

Pode ainda ser abordados um ou mais dos seguintes tópicos adicionais:

- Variedades de riemannianas. Curvatura. Espaços simétricos. Exemplos clássicos.

- Cohomology de Rham, cohomology singular e teorema de Rham.

- Grau de uma aplicação. Índice de um campo de vectores. Aplicações.

- Distribuições. Teoremas de Frobenius e Stefan- Sussmann.

Head Lecturer(s)

António De Nicola

Assessment Methods

Assessment
Frequency: 33.0%
Exam: 67.0%

Bibliography

D. Barden and C. Thomas, An Introduction to Differential Manifolds, Imperial College Press, London, 2003.

J. Lafontaine, An Introduction to Differential Manifolds, Springer, 2015.

M.W. Hirsch, Differential Topology, Corrected reprint of the 1976 original, Graduate Texts in Mathematics, vol. 33,   Springer-Verlag, New York, 1994.

J.M. Lee, Riemannian manifolds: An introduction to Curvature, Graduate Texts in Mathematics, vol. 176, Springer-Verlag, New York, 1997.

J. W. Milnor, Topology from the differentiable viewpoint, Based on notes by David W. Weaver, Revised reprint of the 1965 original, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997.