TOPICS IN SUB RIEMANNIAN GEOMETRY, PADOVA - 2020/21. (DAVIDE BARILARI 1st part, ANDREI AGRACHEV 2nd part)

This is a 30h course for Scuola Galileiana and PhD in Mathematics, University of Padova, academic year 2020/21, on Zoom.
The 1st part (D.Barilari) is introductory to the subject and covers basic and some fundamental results.
The 2nd part (A.Agrachev) is focused on more advanced questions on abnormal extremals and length-minimizers.

For UNIPD users: you can subscribe to the MediaSpace UNIPD page HERE
For NON UNIPD users: please find link to videos HERE or below

Link to the draft version of the book A Comprehensive Introduction to Sub-Riemannian Geometry
Link to the draft version of the lecture notes Differential Geometry course 2022/23 (for Master students at UNIPD)

If you are interested in the Zoom Link, please write me an email.

Lectures 1st part (Davide Barilari)

• Lecture 1. Introduction: Parking a car, rolling a ball [notes] [video]
• Lecture 2. Vector fields, Lie brackets, Frobenius theorem [notes] [video]
• Lecture 3. Sub-Riemannian structures and Rashevski-Chow theorem [notes] [video]
• Lecture 4. Existence of length-minimizers. The Heisenberg group . Length vs Energy [notes] [video]
• Lecture 5. First-order conditions. Normal and abnormal length-minimizers. [notes] [video]
• Lecture 6. Hamiltonian formulation. Abnormal length-minimizers in dimension three. [notes] [video]
• Lecture 7. Normal length-minimizers. The Heisenberg group (continuation). [notes] [video]

Lectures 2st part (Andrei Agrachev)

• Lecture 8. Evaluation map, end-point map and abnormal extremals. [notes] [video]
• Lecture 9. Examples. Isoperimetric problems. Symplectic geometry. [notes] [video]
• Lecture 10. Symplectic characterization of abnormal extremals. Characteristic variety. [notes] [video]
• Lecture 11. Characteristic variety (continuation). Genericity conditions. [notes] [video]
• Lecture 12. Rank 2 distributions. Engel and Cartan distributions. [notes] [video]
• Lecture 13. Abnormals in dimension three. Generic cases and the Sard conjecture. [notes] [video]
• Lecture 14. Rigidity and abnormal curves. Some second order conditions. [notes] [video]
• Lecture 15. Goh and Generalized Legendre conditions. [notes] [video]

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