In 2007-2008 I’ll give an introductory Master course (36h of lectures) on diferential geometry and subRiemannian geometry. This page is used for my preparation of the course.
Topics that I want to cover (on sub-Riemannian geometry)
* Non-integrable distributions and their normal forms ?
* Sub-Riemannian metric and sub-Riemannian distance
* Chow theorem: bracket generating condition –> any two points can be connected by a horizontal path
*Existence of geodesics ?
* Nonholonomic derivatives, nonholonomic degree of functions and vector fields
* Growth vector, privileged coordinates
* Ball-Box theorem
* Nilpotent approximation (homogenization), tangent metric spaces
* Hausdorff dimension and Hausdorff measure
* Singular horizontal curves, abnormal geodesics
* Applications and related topics:� control theory ?� holomorphic functions ? Kaluza-Klein theory ? isoperimetric problems ? geometric phases in mechanics ? hyperbolic groups ?
Recommended References :
* F. Jean, Sub-Riemannian Geometry, notes of the lectures given in Trieste, 2003. [Very good introductory notes, though unfortunately they are incomplete]
* R. Montgomery, A tour of Riemannian geometry, 2001. [A nice book about subRiemannian geometry and some of its applications. Warning: the proof of the ball-box theorem in this book is not correct]