Singular metrics and associated conformal groups underlying ...

12 K. R. Payne
where ∆˜g is Laplace-Beltrami operator on N (cf. [4] and the references
therein). Here the principal part is also singular, as opposed to the smoothness
that (3.2) has.
In a singular geometric situation described by degeneration in the bilinear
form γ on T ∗ M, one can try to study the geodesics on M (constant
velocity locally extremal curves with respect to some length functional) via
a suitable Hamiltonian system (the bicharacteristics of the quadratic form
defined by γ) as opposed to the Lagrangian approach on the tangent bundle
T M, which becomes singular at the degeneracies of γ. In regular (semi-
Riemannian) situations, the two approaches coincide in the sense that up to
reparametrization, the geodesics are projections onto M of the bicharacteristics.
More importantly, under favorable circumstances, the same continues
to hold with respect to a given underlying (singular) geometry. This is the
case for classes of sub-Riemannian manifolds (see section 3.4 of [2] and the
references therein). Such classes contain some of our degenerate elliptic examples
as will be clarified below. On the other hand, for all of the metrics we
consider, one can calculate explicitly and globally the bicharacteristics which
then describe the geodesics away from the metric singularity. In the singular
Lorentzian and Riemannian-Lorentzian cases, one can study global causality
properties such as disprisoning for null geodesics and null pseudo-convexity
(cf. [1] ) as we have done for singular mixed signature metrics of the form
ds 2 = dy 2 + N
j,k=1 (yA)−1
jk dxjdxk, with A(x, y) smooth and positive definite
(cf. [21] and [20]).
Finally, we note the connection of our examples to the notion of a sub-
Riemannian geometry on an n dimensional manifold M in which a metric is
associated to a family of m ≤ n smooth vector fields. For example, for an
operator L (1.1) of Grushin type with K(y) = y 2k , k ∈ N one can write L
as a sum of squares of vector fields L = N+1
j=1 (Xj ) 2 with X j = y k Dj for
1 ≤ j ≤ N and X N+1 = DN+1. These vector fields together with their Lie
brackets generate T R N+1 and hence there is a natural Carnot-Carathéodory
metric associated to this family of vector fields by using piecewise integral
curves of ±X j (see [2], [10] for example) which defines the sub-Riemannian
structure. Notice that the Grushin system of vector fields is not of constant
rank, as is often assumed, and that the geometry is truly sub-Riemannian
only along the degenerate hypersurface Σ (it is Riemannian away from Σ).
This relaxation on the rank is essential for certain applications to geometric
control theory and hypoelliptic equations as is pointed out by Bellaïche
[2]. Carnot-Carathéodory metrics can often be given a meaning even with
low order regularity in the coefficients which has implications for degenerate
elliptic equations such as Harnack inequalities and Hölder regularity results
[6], [7]. Moreover, with respect to the natural Sobolev spaces defined by the
vector fields, one recovers critical exponent phenomena for degenerate and
mixed type equations like those well known in the elliptic setting (cf. [8],
[14]). As pointed out by Strichartz [22], it also makes sense to speak about a
sub-Lorentzian geometry associated to any non-degenerate quadratic form on
a sub-bundle (constant rank) S of the tangent bundle with the bracket generating
property. Such geometries have been studied recently [9] under the
constant rank assumption. We do not know if a non-constant rank version of