Differential geometry

2022/2023
Programme:
Mathematics, Second Cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M3
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Core topics:
Introduction: Vector fields and Lie bracket. Fundamental notions of the Lie theory. Differential forms. Vector bundles, Riemann structures on vector bundles.
Principal bundles, associated bundles, frame bundles, reductions of bundles.
Differential forms with values in Lie algebras, connections on principal bundles. Horizontal lift of a path. Curvature and holonomy. Various descriptions of the curvature on a principal bundle.
Connections on vector bundles, covariant derivative. Chern classes.
Fundamental notions of Riemann geometry: Riemannian metric, Levi-Civitá connection, Riemann curvature tensor and its properties, Ricci and Weyl curvatures, autoparallel curves, geodesic curves. Exponential map.
Additional topics: Subgroups of the group GL(n, C) and symmetric spaces. Gaussian curvature on surfaces. Poisson and symplectic manifolds. Pontryagin classes and Bott's theorem. Conformality and Weyl tensor

Readings

B. A. Dubrovin, A. T. Fomenko, S. P. Novikov: Modern Geometry - Methods and Applications II : The Geometry and Topology of Manifolds, Springer, New York, 1985.
S. Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces, AMS, Providence, 2001.
S. Kobayashi, K. Nomizu: Foundations of Differential Geometry I, II, John Wiley & Sons, New York, 1996.
P. Petersen: Riemannian Geometry, Springer, New York, 1997.
J. Cheeger, D. Ebin, Comparison Theorems in Riemannian Geometry, AMS Chelsea Publishing, Providence, 2008

Objectives and competences

Fundamental concepts of modern differential geometry are introduced. The central objects of the course are connections on principal or vector bundles and their curvatures. The curvature is described from the point of view of the Frobenius theorem. The notion of holonomy is introduced and the relationship between holonomy and curvature is described. These notions are then used in the presentation of the fundamentals of the Riemannian geometry. The relationship between differential geometry and topology is illustrated by means of Chern classes.

Intended learning outcomes

Knowledge and understanding: Understanding the fundamental definitions and concepts of differential geometry.
Application: Solving problems by applying the relevant theory.
Reflection: Understanding the theory through its applications.
Transferable skills: Skills in the use of the relevant literature and other sources, formulating problems and solving them, critical analysis.

Learning and teaching methods

Lectures, exercises, homework, consultations

Assessment

Written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Janez Mrčun:
MRČUN, Janez. An extension of the Reeb stability theorem. Topology and its Applications, ISSN 0166-8641. [Print ed.], 1996, let. 70, št. 1, str. 25-55. [COBISS-SI-ID 7630169]
MOERDIJK, Ieke, MRČUN, Janez. On integrability of infinitesimal actions. American journal of mathematics, ISSN 0002-9327, 2002, vol. 124, no. 3, str. 567-593. [COBISS-SI-ID 11700057]
MOERDIJK, Ieke, MRČUN, Janez. Introduction to foliations and Lie groupoids, (Cambridge studies in advanced mathematics, 91). Cambridge, UK: Cambridge University Press, 2003. IX, 173 str., ilustr. ISBN 0-521-83197-0. [COBISS-SI-ID 12683097]
Pavle Saksida:
SAKSIDA, Pavle. Nahm's equations and generalizations Neumann system. Proceedings of the London Mathematical Society, ISSN 0024-6115, 1999, let. 78, št. 3, str. 701-720. [COBISS-SI-ID 8853849]
SAKSIDA, Pavle. Integrable anharmonic oscillators on spheres and hyperbolic spaces. Nonlinearity, ISSN 0951-7715, 2001, vol. 14, no. 5, str. 977-994. [COBISS-SI-ID 10942809]
SAKSIDA, Pavle. Lattices of Neumann oscillators and Maxwell-Bloch equations. Nonlinearity, ISSN 0951-7715, 2006, vol. 19, no. 3, str. 747-768. [COBISS-SI-ID 13932377]
Sašo Strle:
STEFANOVSKA, Aneta, STRLE, Sašo, KROŠELJ, Peter. On the overestimation of the correlation dimension. Physics letters. Section A, ISSN 0375-9601. [Print ed.], 1997, vol. 235, str. 24-30. [COBISS-SI-ID 607828]
RUBERMAN, Daniel, STRLE, Sašo. Mod 2 Seiberg-Witten invariants of homology tori. Mathematical research letters, ISSN 1073-2780, 2000, vol. 7, no. 5-6, str. 789-799. [COBISS-SI-ID 10557785]
STRLE, Sašo. Bounds on genus and geometric intersections from cylindrical end moduli spaces. Journal of differential geometry, ISSN 0022-040X, 2003, vol. 65, no. 3, str. 469-511. [COBISS-SI-ID 13135193]