Open-dense orbit theorem
A. Zeghib
Speaker: A. Zeghib
Abstract: For a geometric structure on a space M, we consider P its pseudo-group of local symmetries, i.e. all diffeomorphisms between open subsets of M preserving this structure.
When P has a dense orbit, all (continuous) scalar invariants of the structure will be constant. This is weaker than the fact that M being locally homogeneous, in the sense that P has a single orbit which is the whole M.
The open-dense orbit theorem, due to Gromov, says that this essentially is the case when the geometric structure is rigid, e.g. a Riemannian metric or a connection. More precisely, for such a structure, if the pseudo-group of local symmetry admits a dense orbit, then it is open! We present a proof of this theorem using ideas and concepts of control theory and real algebraic geometry.
Lecture notes avaliable here.