10 WENSHENG LIU AND HECTOR J. SUSSMANN
product G(x). The geodesic equations become x = ^ , A = ^ , i.e.
the usual Hamilton equations, whose solutions are the bicharacteris-
tics of H. The constraint
||X||G
=
1 becomes the equality
||A||G
= 1,
i.e. H = | . The projections on M of the bicharacteristics are the
characteristics of H. In other words, Formula (1) associates to the
Riemannian metric G a smooth real-valued function H on the cotan-
gent bundle T*M. The symplectic structure ofT*M associates to H
a Hamiltonian vector field H. The projections on M of the integral
curves of H along which H = | are the geodesies parametrized by
arc-length. More generally, the geodesies parametrized by a constant
times arc-length are precisely the projections on M of arbitrary non-
null bicharacteristics of H (i.e. integral curves of H along which the
constant value of H is ^ 0).
2.2 The Heisenberg algebra case
We now discuss the simplest example of a sub-Riemannian structure
which is not Riemannian. Let UJ be the 1-form in
]R3
given by
UJ
= dz (xdy y dx) (2)
We wish to consider arcs 7 in R
3
that satisfy the velocity constraint
(a;, 7) = 0. Precisely, we let Ca^ be the set of all absolutely continuous
curves 7 : [a, b]
M3
that satisfy (uj(^(t))^{t)) = 0 for almost all
t G [a, 6], and define C U_00a^+00Ca?6. Equivalently, if we let
E(q) ker uo{q) for q G R
3
, then E(q) is a two-dimensional subspace
of R
3
for each q (so E is a ''two-dimensional distribution" in R
3
) , and
the curves 7 in C are those that satisfy j(t) G £"(7(t)) for almost all t.
It is not hard to see that, in spite of the constraint ^y(t) G E(j(t))^
every pair g1? q2 of points of R
3
can be joined by a curve in C. (For
example, the two vector fields / = |^ y^ and g = | - + x|^ form a
basis of sections of E. The Lie bracket [f,g] is equal to 2|^, so / , g
and [/. g] are linearly independent at each point. It then follows from
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