# Some geometric constructions of holonomy plane fields and their analysis. IHP Workshop "Equivalence, invariants, and symmetries of vector distributions and related structures: from Cartan to Tanaka and beyond". Institut Henri Poincaré, Paris France. December 10, 2014

Invited Talk

As is well-known, the holonomic system of a ball rolling without slipping or twisting on a plane is described by a 2-plane field with growth vector (2,3,5), but this is not the famous plane field discovered by Cartan and Engel, whose local automorphism group is of dimension 14. However, the case of a ball of radius 1 rolling without slipping or twisting over a ball of radius 3 does turn out to be locally isomorphic to the Cartan/Engel 2-plane field. Recently, Nurowski and An found examples of surfaces whose holonomic system when rolling over a plane *is* the Cartan/Engel 2-plane field, and I have shown that, for one surface rolling over another (where the two surfaces have distinct Gauss curvatures), the associated 2-plane field cannot be of Cartan/Engel type unless at least one of the surfaces has constant Gauss curvature. I will report on this result and some other related geometric constructions of 2- and 3-plane fields whose equivalence with the flat model is of interest.

### Service Performed By

- Bryant, Robert Phillip Griffiths Professor of Mathematics

### Date

- December 10, 2014

### Service or Event Name

- IHP Workshop "Equivalence, invariants, and symmetries of vector distributions and related structures: from Cartan to Tanaka and beyond"

### Host Organization

- Institut Henri Poincaré, Paris France

### Location or Venue

- Institut Henri Poincaré, Paris France