UNIVERSITAT POLITÈCNICA de CATALUNYA
FACULTAT de MATEMÀTIQUES i ESTADÍSTICA
34966 Differentiable manifolds
Master in Advanced Mathematics and Mathematical Engineering
Xavier Gràcia (coordinator)
The core of this course is devoted to the study some topics in
differential geometry, with particular attention to Lie groups
Review on manifolds
Manifolds, tangent vectors, tangent bundle, submanifolds
Differential equations, tensor fields, Lie derivative, tangent subbundles
Connections and riemannian manifolds
Fundamental group and covering spaces
De Rham cohomology
Lie groups and Lie algebras
Actions of Lie groups on manifolds
Relation between Lie groups and Lie algebras
Symmetries of differential equations
students should have had a basic course on smooth manifolds,
as for instance the one
So, they are expected to be familiar with
tangent and cotangent vectors,
tangent bundle and vector fields,
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
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
John M. Lee
Introduction to smooth manifolds
John M. Lee
Riemannian manifolds: an introduction to curvature
Introduction aux variétés différentielles
(Presses Universitaires de Grenoble, 1996)
Frank W. Warner
Foundations of differentiable manifolds and Lie groups
William M. Boothby
An introduction to differentiable manifolds and riemannian geometry
(Academic Press, 1986)
Loring W. Tu
An introduction to manifolds
(Gauthier-Villars, 1968, 1970, 1971, 1975)
Ivan Kolár, Peter W. Michor, Jan Slovák
Natural operations in differential geometry
Robert H. Wasserman
Tensors and manifolds with applications to physics,
(Oxford University Press, 2004)
José F. Cariñena
Introducción a la geometría diferencial
(lecture notes by Prof. Cariñena, University of Saragossa)
Werner Greub, Stephen Halperin, Ray Vanstone
Connections, curvature, and cohomology,
(Academic Press, 1972-1973)
Paulette Libermann, Charles-Michel Marle
Symplectic geometry and analytical mechanics
(D. Reidel, 1987)
Ana Cannas da Silva
Lectures on symplectic geometry
Algebraic topology and topological algebra
Lev S. Pontrjagin
Topological groups or
Topologie générale (chap. 3)
Groupes de Lie et algèbres de Lie (chap. 1)
William S. Massey
Algebraic topology: an introduction
Raoul Bott, Loring W. Tu
Differential forms in algebraic topology
Lie groups and Lie algebras
Introduction à la théorie des groupes de Lie
Johannes J. Duistermaat, Johan A.C. Kolk
Mikhail M. Postnikov
Lie groups and Lie algebras (Lectures in geometry V)
Melvin Hausner and Jacob T. Schwartz
Lie groups, Lie algebras
(Gordon and Breach, 1968)
Arkadii L. Onishchik, Ernest B. Vinberg
Lie groups and algebraic groups
Differential geometry, Lie groups, and symmetric spaces
(American Mathematical Society, 1978/2001).
Hossein Abbaspour, Martin Moskowitz
Basic Lie theory
(World Scientific, 2007)
Applications of Lie groups
Mark A. Naimark, A.I. Stern
Theory of group representations
Theodor Bröcker, Tammo tom Dieck
Representations of compact Lie groups
Veeravalli S. Varadarajan
Lie groups, Lie algebras, and their representations
David H. Sattinger, Oliver L. Weaver
Lie groups and algebras with applications to physics, geometry, and mechanics
Peter J. Olver
Applications of Lie groups to differential equations 2nd ed.
L. È. Èl'sgol'c
Differentsial'nye uravneniya i variatsionnoe ischislenie (1965)
Differential equations and the calculus of variations (1970)
Applications élémentaires au calcul des variations et à
la théorie des courbes et des surfaces (1967)
Jürgen Jost, Xianqing Li-Jost
Calculus of variations (1998)
Vladimir I. Arnol'd
Matematicheskie metody klassicheskoĭ mekhaniki
Mathematical methods of classical mechanics (1974/1989)
Jorge V. José, Eugene J. Saletan
Classical dynamics. A contemporary approach (1998)
Ralph Abraham, Jerrold E. Marsden
Foundations of mechanics (1978)
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Created on 13 February 2012.
Updated on 18 July 2016.