Vorlesung

D. Kotschick: Foliations and Subriemannian Geometry

Time and place: Wed 11-13, Thu 16-18, room E 05
Recitation classes: Thu 9-11, room E 46
Contents: Subriemannian geometry is the Riemannian geometry (metrics, connections, curvature) of subbundles of the tangent bundle. The origin of this subject is in physics, e.g. nonholonomic mechanics and control theory, but our treatment will be mathematical. We shall formulate constraints and accessability as integrability conditions on distributions, and study the differential geometry of distributions. Some of the important relations with symplectic and contact geometry and with the theory of integrable Hamiltonian systems will be discussed. For integrable distributions, i. e. foliations, we shall study the conditions on a foliated manifold to admit Riemannian metrics which make the leaves minimal or totally geodesic.
Syllabus
Intended audience: Diplom, Master and doctoral students in mathematics and/or physics.
Prerequisites: Some previous exposure to differential geometry and/or topology.
Introduction:
M. H. Freedman and F. Luo: Selected Applications of Geometry to Low-Dimensional Topology. Amer. Math. Soc. 1989.
General reference for foliations:
C. Godbillon: Feuilletages. Birkhäuser Verlag 1991.

Specialised books:

A tour of subriemannian geometries, their geodesics and applications.

Foliations on Riemannian manifolds and submanifolds.

Riemannian Foliations.