Part I. Motivation and problem formulation
In the usual formulation of nonlinear control theory, one has a drift vector field, representing the uncontrolled dynamics, and control vector fields, representing that part of the dynamics that one can control. The choice of drift vector field and control vector fields is not unique, and therefore a formulation is developed that is independent of these choices. Very little work has been done on this purely geometric formulation of control theory. To give some idea of the sorts of problems that arise, we formulate some novel definitions of controllability.

Part II. Controllability theorems
Carrying on from part I of the talk, some results describing controllability are developed. ``Zeroth-order'' and ``first-order'' results from the literature are given in the geometric formulation. Finally, new second-order controllability results are stated.

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