# Local geometry of nonregular quasimetric "Carnot-Carathéodory spaces"

### (Sobolev Institute of Mathematics)

**Event:** ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

**Date:** Oct 11, 2011,
**time:** 16:10

**Place:** Centro De Giorgi, Scuola Normale Superiore

**Abstract.** Carnot-Carathéodory spaces are a wide generalization of sub-Riemannian

manifolds which model nonholonomic processes and naturally arise in many applications. We consider the case of arbitrary weighted filtration of the tangent bundle (it generalizes the sub-Riemannian framework of a bracket-generating distribution) and study the local geometry of such spaces in a neighborhood of nonregular points (where dimensions of the subbundles generating the filtration may vary from point to point). In particular, we prove analogs of such classical results of sub-Riemannian geometry as Local approximation theorem and Tangent cone theorem. Quasimetrics are needed since, in the considered situation, the intrinsic Carnot-Carathéodory metric might not exist. The motivation of this research stems from nonlinear control theory and subelliptic equations.