M7110 Differential Geometry

Faculty of Science
Autumn 2010 - only for the accreditation
Extent and Intensity
2/2. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Recommended Type of Completion: zk (examination). Other types of completion: z (credit).
doc. Lukáš Vokřínek, PhD. (lecturer)
Guaranteed by
doc. RNDr. Martin Čadek, CSc.
Department of Mathematics and Statistics - Departments - Faculty of Science
M5130 Global Analysis
Before enrolling the course the students should pass "Differential Geometry of Curves and Surfaces" and "Global Analysis".
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The course presents the basic knowledge of contemporary differential geometry that prepares the student for independent reading of mathematical literature.

At the end of the course students should be able to:
*explain the connection between Lie groups and their Lie algebras
*derive the Lie bracket of the classical (and also not so classical) Lie groups
*understand the relationship between the principal and associated bundles and connections on them
*understand the relationship between different forms of connection and their impact on the geometry of a Riemannian space
  • Lie groups and Lie algebras. Actions of Lie groups on manifolds. Vector bundles and fibered manifolds. Principal and associated bundles. Connections on principal bundles, parallel transport. Linear connections on vector bundles. Koszul's approach to connections on tangent bundles. Riemannian metric and the Levi-Civita connection. Some applications.
  • KOLÁŘ, Ivan, Jan SLOVÁK and Peter W. MICHOR. Natural Operations in Differential Geometry. Berlin-Heidelberg-New York: Springer-Verlag, 1993. 434 pp. ISBN 3-540-56235-4. info
  • Sharpe, Richard W. Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer, 1997
Teaching methods
Lectures and tutorials.
Assessment methods
An oral exam.
Language of instruction
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is taught: every week.
General note: podzim 2006 jako konzultovaná četba.
The course is also listed under the following terms Autumn 1999, Autumn 2000, Autumn 2001, Autumn 2003, Autumn 2006, Autumn 2008, Autumn 2010, Autumn 2011 - acreditation, Autumn 2012, Autumn 2014, Autumn 2016, Autumn 2018, Spring 2021.