UNIVERSITAT POLITÈCNICA de CATALUNYA
•
FACULTAT de MATEMÀTIQUES i ESTADÍSTICA
34966 Differentiable manifolds
Master in Advanced Mathematics and Mathematical Engineering
2015 spring
Professors:
Xavier Gràcia (coordinator)
Contents
The core of this course is devoted to the study some topics in
differential geometry, with particular attention to Lie groups
and applications.

Review on manifolds

Manifolds, tangent vectors, tangent bundle, submanifolds

Differential equations, tensor fields, Lie derivative, tangent subbundles

Differentialgeometric structures

Connections and riemannian manifolds

Symplectic manifolds

Vector bundles

Complements

Variational calculus

Fundamental group and covering spaces

Topological groups

De Rham cohomology

Lie groups and Lie algebras

Lie groups

Actions of Lie groups on manifolds

Lie algebras

Relation between Lie groups and Lie algebras

Some applications

Analytical mechanics

Symmetries of differential equations
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.
Schedule
The timetable is
Monday 17:3019:30
and
Friday 1517,
room 101
of the FME.
Regular sessions will take place from 9 February to 22 May, 2015.
Evaluation
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.
Problems
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.
Essays
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 1520 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.
Bibliography
Differential geometry (including riemannian geometry and
Lie groups)

John M. Lee
Introduction to smooth manifolds
2nd ed.
(Springer, 2013)

John M. Lee
Riemannian manifolds: an introduction to curvature
(Springer, 1997)

Lawrence Conlon
Differentiable manifolds
2nd ed.
(Birkhäuser, 2003)

Jacques Lafontaine
Introduction aux variétés différentielles
(Presses Universitaires de Grenoble, 1996)

Frank W. Warner
Foundations of differentiable manifolds and Lie groups
(Springer, 1971/1983)

William M. Boothby
An introduction to differentiable manifolds and riemannian geometry
2nd ed.
(Academic Press, 1986)

Loring W. Tu
An introduction to manifolds
(Springer, 2008)

Jean Dieudonné
Éléments d'analyse,
vols. 25
(GauthierVillars, 1968, 1970, 1971, 1975)

Ivan Kolár, Peter W. Michor, Jan Slovák
Natural operations in differential geometry
(Springer, 1993)

Robert H. Wasserman
Tensors and manifolds with applications to physics,
2nd ed.
(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,
vols. 12
(Academic Press, 19721973)

Paulette Libermann, CharlesMichel Marle
Symplectic geometry and analytical mechanics
(D. Reidel, 1987)

Ana Cannas da Silva
Lectures on symplectic geometry
(Springer, 2001)
Algebraic topology and topological algebra

Lev S. Pontrjagin
Topological groups or
Grupos continuos

Nicolas Bourbaki
Topologie générale (chap. 3)
(Hermann, 1971)

Nicolas Bourbaki
Groupes de Lie et algèbres de Lie (chap. 1)
(Hermann, 1971)

William S. Massey
Algebraic topology: an introduction
(Springer, 1977)

Raoul Bott, Loring W. Tu
Differential forms in algebraic topology
(Springer, 1982)
Lie groups and Lie algebras

Roger Godement
Introduction à la théorie des groupes de Lie
(Springer, 2004)

Johannes J. Duistermaat, Johan A.C. Kolk
Lie groups
(Springer, 2000)

Mikhail M. Postnikov
Lie groups and Lie algebras (Lectures in geometry V)
(Mir, 1986)

Nathan Jacobson
Lie algebras
(Interscience, 1962)

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
(Springer, 1990)

Sigurdur Helgason
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
(Springer, 1982)

Theodor Bröcker, Tammo tom Dieck
Representations of compact Lie groups
(Springer, 1995)

Veeravalli S. Varadarajan
Lie groups, Lie algebras, and their representations
(Springer, 1974/1984)

David H. Sattinger, Oliver L. Weaver
Lie groups and algebras with applications to physics, geometry, and mechanics
(Springer, 1986/1993)

Peter J. Olver
Applications of Lie groups to differential equations 2nd ed.
(Springer, 1993)
Related topics

L. È. Èl'sgol'c
Differentsial'nye uravneniya i variatsionnoe ischislenie (1965)
Differential equations and the calculus of variations (1970)

Henri Cartan
Formes différentielles.
Applications élémentaires au calcul des variations et à
la théorie des courbes et des surfaces (1967)

Jürgen Jost, Xianqing LiJost
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)
Other informations

Back to Xavier Gràcia's homepage

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