34966 Differentiable manifolds

2015 spring

Contents

• Review on manifolds
  • Manifolds, tangent vectors, tangent bundle, submanifolds
  • Differential equations, tensor fields, Lie derivative, tangent subbundles
• Differential-geometric 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

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

Evaluation

Problems

Essays

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.

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. 2-5 (Gauthier-Villars, 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. 1-2 (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 (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 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)

Other informations

• Description of the course according to the teaching guide:
  local copy, retrieved from the list of courses of the Master on 9 Feb 2015
  link to the web of the FME.

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