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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 di erential 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 di erential on the curve. The so-called 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, cross-ratio.
1 Introduction
Suppose M is a smooth n-dimensional 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
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