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