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by Lê Đăng Doanh (thuyết trình trước Bộ Chính Trị)

SubRiemannian

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]

* …

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