 
Summary: Geometry of Jacobi Curves I
Andrei Agrachev Igor Zelenko y
Abstract
Jacobi curves are far going generalizations of the spaces of \Jacobi elds" along Rieman
nian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. In our
paper we develop dierential geometry of these curves which provides basic feedback or gauge
invariants for a wide class of smooth control systems and geometric structures. Two principal
invariants are: the generalized Ricci curvature, which is an invariant of the parametrized curve
in the Lagrange Grassmannian providing the curve with a natural projective structure, and a
fundamental form, which is a degree 4 dierential on the curve. The socalled rank 1 curves
are studied in greater detail. Jacobi curves of this class are associated to systems with scalar
controls and to rank 2 vector distributions.
In the forthcoming second part of the paper we will present the comparison theorems
(i.e., the estimates for the conjugate points in terms of our invariants) for rank 1 curves and
introduce an important class of \
at curves\.
Key words: Lagrange Grassmannian, Jacobi curve, symplectic invariants, feedback invari
ants, crossratio.
1 Introduction
Suppose M is a smooth ndimensional manifold and : T M !M is the cotangent bundle to
M: Let H be a codimension 1 submanifold in T M such that H is transversal to T
