2018/2019
Differential Geometry
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Optional course (faculty)
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
3, 4 module
Language:
English
ECTS credits:
5
Contact hours:
76
Course Syllabus
Abstract
The course will serve as an introductory guide to basic topics of Differential geometry: very first introduction to Symplectic and Contact Geometry, the theory of Riemannian manifolds, the theory of affine connections on manifolds, geodesics.
Learning Objectives
- Students will be introduced to the very basic notions of classical differential geometry and then will be given the precise examples of applying characteristic classes, connections and curvature tensors on Riemmanian manifolds.
Expected Learning Outcomes
- Students would get familiar with with basic notions and instruments of differential geometry, would enhance their methods of solving mathematical problems in various fields. Students would be capable of solving basic problems on differential geometry structures and objects on manifolds.
Course Contents
- Symplectic and Contact structures. Darboux theorems. Reductions.
- Differential connection.
- Parallel transport. Curvature.
- Affine connection.
- Introduction to characteristic classes.
- Riemannian manifold. Levi–Civita connection.
- Riemannian curvature tensor.
- Geodesics. The Hopf–Rinow theorem.
- First and second variation of arc length.
- Jacobi’s equation and conjugate points.
Bibliography
Recommended Core Bibliography
- John Milnor. (2016). Morse Theory. (AM-51), Volume 51. Princeton: Princeton University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1179997
Recommended Additional Bibliography
- Arnold Neumaier, & Dennis Westra. (2011). Classical and quantum mechanics via Lie algebras. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.D337675E
- Neumaier, A., & Westra, D. (2008). Classical and Quantum Mechanics via Lie algebras. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.0810.1019