PRESENCE im SoSe22: Differential Geometry in the Method of Cartan Geometry - Einzelansicht

Topics covered in this course

Frobenius theorem, Cartan geometry, basic definitions in parabolic geometry such as regularity, examples such as projective, conformal, and parabolic contact geometries.

Textbooks and References

Primary reference: Differential Geometry, Cartan's Generalization of Klein's Erlangen Program by R. W. Sharpe. & Parabolic Geometries I by Cap & Slovak

Additional references: Differential Geometric structure by W. Poor & Lie Group beyond introduction by Knapp. Cartan for Beginners by Ivey and Landsberg may also be helpful.

Tentative Schedule of the course

Week 1: Review on the basic definitions differential forms, vector bundles, principal bundles, associated bundles

Week 2: Connections, covariant derivatives, and curvatures.

Week 3: Foliations and Frobenius theorem.

Week 4-5: Review on basic Lie group theory, decomposition of semi-simple Lie algebras

Week 6: Fundamental theorem of calculus and the structure equations.

Week 7: Basic definitions in Cartan geometry, i.e. Cartan connections, curvatures and gauges.

Week 8: First basic example: Cartan bundles of Riemannian geometry.

Week 9-10: Introduction to parabolic geometries. Regularity and Normality.

Week 11: Development map and distinguished curves.

Week 12-13: Examples in [1]-graded geometries: projective and conformal.

Week 14: Examples in [2]-graded geometries: parabolic contact geometry.