34966 Differentiable manifolds

Master in Advanced Mathematics and Mathematical Engineering

2015 spring

Professors: Xavier Gràcia (coordinator)


The core of this course is devoted to the study some topics in differential geometry, with particular attention to Lie groups and applications.  

Previous knowledge
Preferably students should have had a basic course on smooth manifolds, as for instance the one here.   So, they are expected to be familiar with manifolds, tangent and cotangent vectors, tangent bundle and vector fields, submanifolds, differential equations on manifolds, tensor fields and differential forms, and Lie derivatives.   These topics are covered by many books, as for instance Lee, Lafontaine, Conlon, Boothby, Warner, ..., as well as in several course notes available in the WWW.   However, they will reviewed at the beginning of the course.


The timetable is Monday 17:30-19:30 and Friday 15-17, room 101 of the FME.

Regular sessions will take place from 9 February to 22 May, 2015.


Evaluation is based on students' participation and homework (exercises and problems), and on the completion and presentation of an essay (a written work) on a topic on differential geometry. Eventually, there will be a final examination.


Along the course some proofs are left as exercices, and several collections of problems are assigned.  The students are expected to solve and deliver several of them, and occasionally to present them on the blackboard.


Presentation of essays takes place by the end of the academic year.  Students should send me a preliminary version of the file the day before their presentation.  The final version is due on Monday 15 June.

About the essays
A typical essay may have about 15-20 pages.  It has to be clearly identified (title, author, date, data of the course), and its contents clearly organised (table of contents, a detailed introduction, contents, bibliography).  Copypaste is not only discouraged, but forbidden; you should understand what you want to explain, and do this with your own words.  We are aware that English is not our mother tongue, nevertheless you should try to write it correctly.  The essay has to be delivered as a PDF file (or a similar file format).
Each student has about 30 min for the presentation, and is expected to use mainly the blackboard, though a minor usage of the computer+projector is also allowed.


Differential geometry (including riemannian geometry and Lie groups)

Algebraic topology and topological algebra

Lie groups and Lie algebras

Applications of Lie groups

Related topics

Other informations

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Page https://mat-web.upc.edu/people/xavier.gracia/diffman.  Created on 13 February 2012.  Updated on 18 July 2016.