Universität Wien

250096 VO Analysis on Manifolds (2023S)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik
Moodle

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

  • Wednesday 01.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 03.03. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 08.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 10.03. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 15.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 17.03. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 22.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 24.03. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 29.03. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 31.03. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 19.04. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 21.04. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 26.04. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 28.04. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 03.05. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 05.05. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 10.05. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 12.05. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 17.05. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 19.05. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 24.05. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 26.05. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 31.05. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 02.06. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 07.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 09.06. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 14.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 16.06. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 21.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 23.06. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Wednesday 28.06. 08:00 - 09:30 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock
  • Friday 30.06. 13:15 - 14:45 Hörsaal 11 Oskar-Morgenstern-Platz 1 2.Stock

Information

Aims, contents and method of the course

Aims, contents and method of the course
I expect you to have a good grasp of multivariable calculus and linear algebra, including the inverse and inverse function theorem, basic point set topology, and multilinear maps on finite dimensional vector spaces.
— Geometry of charts in Euclidean space (this may be mostly review)
— Abstract manifolds and the gluing lemma
— Tangent bundle and its universal property
— Maps between manifolds and their associated tangent maps
— Special types of maps and their canonical forms
— Whitney embedding theorem
— Action of tangent fields on functions
— Lie bracket
— Vector bundles and the fundamental principle of tensor calculus
— Differential forms and their calculus
— Integration on manifolds
— Stokes’ theorem
We will discuss further topics such as the following depending on the interests of the audience:
— Sard’s theorem
— Elements of de Rham co-homology
— Flows of tangent fields
— Frobenius theorem on integrable distributions
— Derivations on vector bundles

Assessment and permitted materials

A thorough half-hour oral exam

Minimum requirements and assessment criteria

Examination topics

all material covered in class, unless explicitly mentioned otherwise

Reading list

Lecture notes will be made available for large portions of the course material.

I recommend the following textbooks for extra reading.

Guillemin, Victor; Pollack, Alan Differential topology. Reprint of the 1974 original. AMS Chelsea Publishing, Providence, RI, 2010. xviii+224 pp.

Lee, John M. Introduction to smooth manifolds. Second edition. Graduate Texts in Mathematics, 218. Springer, New York, 2013. xvi+708 pp.

Madsen, Ib; Tornehave, Jørgen From calculus to cohomology. de Rham cohomology and characteristic classes. Cambridge University Press, Cambridge, 1997. viii+286 pp.

Milnor, J. Morse theory. Based on lecture notes by M. Spivak and R. Wells Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963 vi+153 pp.

Montiel, Sebastián; Ros, Antonio Curves and surfaces. Second edition. Translated from the 1998 Spanish original by Montiel and edited by Donald Babbitt. Graduate Studies in Mathematics, 69. American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2009. xvi+376 pp.

O'Neill, Barrett Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. xiii+468 pp.

Association in the course directory

MGED

Last modified: Tu 09.04.2024 10:26