Singular metrics and associated conformal groups underlying ...

Annali di matematica pura ed applicata manuscript No.
(will be inserted by the editor)
Kevin R. Payne
Singular metrics and associated
conformal groups underlying
differential operators of mixed and
degenerate types
Received: date
Abstract For partial differential equations of mixed elliptic-hyperbolic and
degenerate types which are the Euler-Lagrange equations for an associated
Lagrangian, we examine an associated metric structure which becomes singular
on the hypersurface where the operator degenerates. In particular, we
show that the “non-trivial part” of the complete symmetry group for the
differential operator (calculated in a previous paper [15]) corresponds to a
group of local conformal transformations with respect to the metric away
from the metric singularity and that the group extends smoothly across the
singular surface. In this way, we define and calculate the conformal group for
these operators as well as for lower order singular perturbations which are
defined naturally by the singular metric.
Keywords Symmetry groups · Mixed type equations · Singular Riemannian-
Lorentzian metrics · Conformal transformations
Mathematics Subject Classification (2000) 35M10 · 58J70 · 53A30
1 Introduction
The purpose of this note is to formulate precisely and to prove various claims
made in a recent paper [15] concerning the association of a group of conformal
transformations to a class of differential operators of mixed-elliptic hyperbolic
or degenerate type. We consider the operator
Work supported by MIUR, Project “Metodi Variazionali ed Equazioni Differenziali
Non Lineari” and MIUR, Project “Metodi Variazionali e Topologici nello Studio di
Fenomeni Non Lineari”.
K. R. Payne
Dipartimento di Matematica “F. Enriques”, Università di Milano, Via Saldini 50,
20133 Milano, Italy
E-mail: payne@mat.unimi.it

2 K. R. Payne
L = K(y)∆x + ∂ 2 y
(1.1)
where (x, y) ∈ R N × R, ∆x is the Laplace operator on R N with N ≥ 1, and
the coefficient K ∈ C 0 (R) satisfies
K(0) = 0 and K(y) = 0 for y = 0. (1.2)
The equation degenerates along the hypersurface y = 0 and our main interest
will be cases in which K yields a change of type; that is, K also satisfies
yK(y) > 0 for y = 0 (1.3)
so that the operator (1.1) is of mixed type (elliptic for y > 0 and hyperbolic
for y < 0). However, much of what will be discussed depends only on the
form of the degeneracy in (1.1) − (1.2).
In the paper [15], we classified the symmetry groups and calculated the
associated conservation laws for the equation Lu = 0, which is the Euler-
Lagrange equation for the Lagrangian
L(y, u, ∇u) = 1
K(y)|∇xu|
2
2 + u 2 y . (1.4)
In fact, the class defined by (1.1) − (1.3) represents the simplest examples
of second order equations of mixed type associated to a Lagrangian with
degeneracy on a hypersurface. As is to be expected, the largest possible
symmetry groups occur when K takes a pure power form
K(y) = y|y| m−1 , m > 0 (1.5)
in the mixed type case, or ±|y| m in the purely elliptic/hyperbolic but degenerate
cases. The operator in (1.1) with (1.5) is known as the Gellerstedt
operator and gives the Tricomi operator when K(y) = y, while the choice
K(y) = y 2 yields the degenerate elliptic Grushin operator. Such operators
arise in many physical and geometrical problems with a particular structure,
such as: transonic fluid flow [3] [16], quantum cosmology [11], and the imbedding
of manifolds with curvature that changes sign [13]. See also section 6 of
[15] for a brief discussion.
If one takes the limiting case, m = 0 in (1.5), one arrives at the Laurentiev-
Bitsadze operator which glues the Laplacian for y > 0 to the D’Alembertian
for y < 0. It is well known that the symmetry groups for these classical
operators are given by the group of conformal transformations with respect
to the Euclidian and Minkowski metrics respectively (cf. [17] and [5]). We will
show that this also holds in a suitably interpreted sense for the mixed type
operators satisfying (1.1) and (1.5) with respect to a suitable singular metric
of mixed Riemannian-Lorentzian signature as announced in [15]. Moreover,
the analogous result holds in the purely elliptic/hyperbolic but degenerate
setting of (1.1) with K(y) = ±|y| m , m > 0.

Conformal groups for operators of mixed and degenerate types 3
2 Symmetry groups for the operator L
We recall briefly the results in [15] on the symmetry groups for the equation
Lu = 0 (2.1)
with L of the form (1.1) − (1.2) and with power type degeneration (1.5) (or
K(y) = ±|y| m ). One has, apart from certain trivial symmetries for these
linear and homogeneous equations, symmetries coming from: 1) translations
in the “space variables” x; 2) rotations in the space variables; 3) certain
anisotropic dilations; 4) inversion with respect to a well chosen hypersurface
(cf. Theorem 2.5 of [15]).
More precisely, the trivial one parameter symmetry groups arise from the
fact that if u solves (2.1) then so does u + ɛβ with β any solution of (2.1)
and ɛ ∈ R. The non trivial symmetries are represented by the fact that if u
solves (2.1) with K(y) = y|y| m−1 , m > 0, then so do
and
uk;ɛ(x, y) = Tk;ɛu(x, y) = u(x − ɛek, y) (2.2)
uj,k;ɛ(x, y) = Rj,k;ɛu(x, y) = u(Aj,k;ɛx, y) (2.3)
uλ(x, y) = Sλu(x, y) = λ −p(m,N)
u λ −(m+2) x, λ −2
y
uk;ɛ(x, y) = Ik;ɛu(x, y) = D −q(N,m)
k,ɛ u
x + ɛdek
,
Dk,ɛ
where ɛ ∈ R (and |ɛ| is small in (2.5)), λ > 0, {ek} N k=1
of R N , Aj,k;ɛ is the ɛ-rotation in the xj − xk plane,
where
and
p(m, N) =
N(m + 2) − 2
2
y
D 2/(m+2)
k,ɛ
(2.4)
(2.5)
is the standard basis
> 0, (2.6)
Dk,ɛ(x, y) = 1 + 2ɛxk + ɛ 2 d(x, y), (2.7)
d(x, y) = |x| 2 +
q(m, N) =
4
(m + 2) 2 y|y|m+1 , (2.8)
N(m + 2) − 2
2(m + 2)
> 0. (2.9)
The same result holds for the degenerate elliptic/hyperbolic cases where
K(y) = ±|y| m where it is enough to replace y|y| m+1 with ±|y| m+2 in (2.8).
The symmetries (2.2) − (2.4) are variational in the sense that they leave
invariant the integral of the Lagrangian (1.4) while the symmetry (2.5) is
a divergence symmetry and is only locally well defined. All yield associated

4 K. R. Payne
conservation laws. The infinitesimal generators of the nontrivial symmetries
are given by the vector fields
v T k = ∂
, k = 1, . . . , N (2.10)
∂xk
v D = (m + 2)x · ∇x + 2y ∂
∂y
v R jk = xk
v I k = −d(x, y) ∂
N(m + 2) − 2
− u
2
∂
, (2.11)
∂u
∂ ∂
− xj , 1 ≤ j < k ≤ N (2.12)
∂xj ∂xk
+ 2xkx · ∇x +
∂xk
4 ∂
xky
m + 2 ∂y
N(m + 2) − 2
− xku
m + 2
∂
,
∂u
k = 1, . . . , N (2.13)
and together with the trivial symmetries generate the complete symmetry
group (cf. Theorem 2.5 of [15]). In particular, we recall that the proof exploits
well known infinitesimal techniques for classifying the infinitesimal generators
v =
N
ξ i (x, y, u) ∂
i=1
∂xi
+ η(x, y, u) ∂
∂
+ ϕ(x, y, u)
∂y ∂u
(2.14)
of symmetries, where v is thought of as a vector field which acts an open
subset M of the 0-jet space, R N+1 × U (0) R N+1 × R (the space of values
for independent and dependent variables) together with the action of their
prolongations onto higher order jet spaces (which includes the values of higher
order derivatives of u). In the smooth coefficient case for L where K(y) =
y m , which is degenerate elliptic for m even and of mixed type for m odd,
the complete theory can be applied globally on all of R N+1 . Tedious but
elementary calculations show that the coefficients of each generator v in (2.14)
satisfies: ξ i = ξ i (x, y) and η = η(x, y) are independent of u; ϕ = α(x, y)u +
β(x, y) is linear in u; and
Lα = Lβ = 0 (2.15)
Lξ i = 2y m αxi , i = 1, . . . , N (2.16)
Lη = 2αy
(2.17)
2y m ξ i xi − 2ym ηy − my m−1 η = 0, i = 1, . . . , N (2.18)
m
y ξ i xj + ξj
xi
= 0, 1 ≤ i < j ≤ N (2.19)
y m ηxi + ξ i y = 0, i = 1, . . . , N. (2.20)
For m ∈ R + \ N, the lack of regularity in the coefficients creates no essential
difficulty as each group action preserves the half-spaces R N+1
± . See the
Appendix of [15] for details.

Conformal groups for operators of mixed and degenerate types 5
3 A singular Riemannian-Lorentzian structure
As we pointed out in [15], there is a natural geometric structure associated
to the operator (1.1); namely the linear operator in (1.1) is associated to
a singular geometry on R N+1 via the metric g which in global coordinates
(x, y) on R N+1 is given by the matrix
[gij] =
−1 K(y) IN×N 0
. (3.1)
0 1
For K smooth and satisfying (1.2)−(1.3), g is a smoothly varying symmetric
and non degenerate bilinear form on the tangent bundles T R N+1
± and hence
is a smooth Riemannian/Lorentzian metric on the upper/lower half spaces
which becomes singular on the hyperplane y = 0. Notice that the inverse
tensor [gij ] to (3.1) on the cotangent bundles T ∗R N+1
± extends continuously
across the metric singularity and is smooth if m is an integer. If one computes
the associated Laplace-Beltrami operator/D’Alembertian in their respective
, one arrives at the geometrically natural operator
half spaces R N+1
±
Lg = divg ◦ grad g = K(y)∆x + ∂ 2 y − NK′ (y)
2K(y) ∂y
(3.2)
as will be shown below. Hence the linear operator in (1.1) is the principal
and everywhere smooth part of this “geometric operator”. Mixed signature
metrics have received some attention in the context of quantum cosmology,
beginning with the Hawking-Hartle no boundary hypothesis which if true
implies that space-time evolved from a Riemannian past into a Lorentzian
present, as well as models for tunnelling effects in quantum gravity (cf. Chapters
3 and 5 of [11]). More recently, and in the spirit of the present observations,
the description of harmonic forms in certain projective surfaces has
been studied via a Hodge system with respect to a singular metric [19].
As stated in the introduction, we would like to establish precisely the connection
between the symmetry groups of the operator (1.1) or (3.2) and the
group of conformal transformations with respect to this underlying singular
geometric structure (3.1). The analysis proceeds along classical lines away
from the metric singularity on the hypersurface y = 0 and the simple form of
the metric allows for a continuation across the singularity in an explicit way.
To obtain the richest possible classes of symmetries and conformal transformations
we will restrict our attention to type change functions of power
type where three possible situations can be treated in the same way; namely
operators of degenerate elliptic, degenerate hyperbolic, or mixed type:
(DE) K(y) = |y| m , m > 0
(DH) K(y) = −|y| m , m > 0
(MT) K(y) = y|y| m−1 , m > 0
To simply notation, we will treat explicitly only the case
K(y) = y m with m ∈ N (3.3)

6 K. R. Payne
which is of the type (DE)/(MT) for m even/odd. Obvious modifications can
be made for the other cases.
In the rest of this section, we recall briefly the main geometric notions
needed to calculate a “conformal group” associated to the singular metric
(3.1); these conformal structures will be analyzed in section 4. In all cases,
for y > 0 and y < 0 the metric g is a smoothly varying symmetric (diagonal)
non degenerate bilinear form of constant index (0 in the elliptic regions (Riemannian
metric) and N in the hyperbolic regions (Lorentzian metric)) and
hence gives a semi-Riemannian metric on each half space (cf. the monograph
of O’Neill [18]). We recall that the index equals the maximal dimension of
a subspace on which the restricted form is negative definite. One has then
the standard notions of a Levi-Civita connection, covariant derivatives of
tensor fields, Christoffel symbols, divergence and gradient (see [18] as well
as Chapter 2 of the monograph of Taylor [23] whose notation we will often
follow). To simplify notation we will denote the coordinates on RN+1 by
(x1, . . . , xN, xN+1) with xN+1 in the place of y and denote the corresponding
basis elements in T RN+1 by Dk = ∂/∂xk, while ∂kf represents the partial
derivative of a scalar valued function f.
Denoting by X (R N+1
± ) the space of vector fields X with smooth coefficients
X = N+1 k=1 XkDk, one has the covariant derivative of X with respect
to Dj given by the vector field
∇Dj X =
N+1
k=1
X k ;jDk where X k ;j := ∂jX k N+1
+
l=1
Γ k
ljX l
(3.4)
and Γ l jk are the Christoffel symbols or connection coefficients with respect to
the Levi-Civita connection ∇ : X (R N+1
± ) × X (R N+1
± ) → X (R N+1
± ). For our
diagonal metrics one has
Γ l jk = 1
2 gll
∂gjl
+
∂xk
∂gkl
−
∂xj
∂gjk
, j, k, l = 1, . . . N + 1 (3.5)
∂xl
which for (3.1) with (3.3) yields
Γ l ⎧
⎪⎨
−m/(2xN+1) k = N + 1, l = j = N + 1
−m/(2xN+1) j = N + 1, l = k = N + 1
jk =
⎪⎩
m/(2x m+1
N+1 ) l = N + 1, j = k = N + 1
0 otherwise
(3.6)
Making use of (3.5) and (3.6) one defines the divergence of a vector field by
divgX :=
=
N+1
j=1
N+1
j=1
X j
;j =
N+1
∂jX j N
+
j=1
j=1
∂jX j − mN
X
2xN+1
N+1
Γ j
N+1,j XN+1
and the gradient of a scalar valued function u as the vector field
(3.7)

Conformal groups for operators of mixed and degenerate types 7
grad gu =
N+1
j=1
X j Dj where X j :=
N+1
k=1
g jk ∂ku (3.8)
and hence the coefficients of grad gu are identifiable with the vector valued
function
∇gu = (x m N+1∂1u, . . . , x m N+1∂Nu, ∂N+1u) (3.9)
from which the claim (3.2) follows.
For these simple metrics, one calculates easily their associated curvatures
which become infinite as one approaches the singular hypersurface. In particular
the scalar curvature S blows up like CN,mx −2
N+1 . In fact, one easily
computes (again using the fact that g is diagonal)
S :=
N+1
j=1
g jj (Ric)jj =
−m(m + 2)N
2x 2 N+1
where the Ricci curvature Ric is given in the usual way in terms of Γ l jk
(Ric)jk =
N+1
l=1
∂lΓ l kj − ∂kΓ l
lj +
N+1
i=1
(3.10)
l
ΓliΓ i kl − Γ l kiΓ i
lj
. (3.11)
Additional comments concerning the notion of singular geometric structures
and relations to other singular metric structures will be given in section 5.
4 Associated conformal transformations
In terms of the geometric data presented in section 3, one can use the classical
infinitesimal techniques in order to describe the isometries and conformal
transformations on the half-spaces R N+1
± . We recall that a vector field X =
N+1 j=1 Xj Dj generates an isometry (its time-t flow F t X preserves g; that is
[F t X ]∗g = g) if and only if it is a Killing field; that is, its coefficients satisfy
the system
(2DefX)jk :=
N+1
l=1
gklX l ;j + gjlX l
;k = 0, j, k = 1, . . . N + 1 (4.1)
where DefX is the deformation tensor of X, which for a diagonal g reduces
to
(2DefX)jk := gkkX k ;j + gjjX j
;k = 0, j, k = 1, . . . N + 1. (4.2)
Moreover, X generates a conformal transformation ([F t X ]∗ g = α(t, x)g for
some scalar valued function α) if and only if X is a conformal Killing field;
that is

8 K. R. Payne
DefX = 1
N + 1 (divgX) g (4.3)
In preparation for the results which follow we introduce the following definition
suited to our limited aims.
Definition 4.1. Let g be a semi-Riemannian metric on a half-space R N+1
± .
The conformal algebra associated to g is the Lie algebra Kg(R N+1
± ) of all
conformal Killing fields X on R N+1
± ; that is, fields with smooth coefficients
that satisfy (4.3).
The fact that Kg(R N+1
± ) forms a Lie algebra for any g with constant index on
a half space follows the standard theory [18], while the possibility of gluing
together two such algebras along the singular hypersurface will depend on
the particular form of g. We begin with a lemma for the high dimensional
cases (N ≥ 2).
Lemma 4.2. Let K be a pure power type change function of the form (3.3)
on a half-space R N+1
± . If N ≥ 2, then the conformal algebra Kg(R N+1
± ) is
(N + 1)(N + 2)/2 dimensional and spanned by the vector fields
Ik = −dDk + 2xk
where
S = (m + 2)
Tk = Dk, k = 1, . . . , N (4.4)
N
(xjDj) + 2xN+1DN+1
j=1
(4.5)
Rjk = xkDj − xjDk, 1 ≤ j < k ≤ N (4.6)
N
(xjDj) + 4
m + 2 xkxN+1DN+1, k = 1, . . . , N (4.7)
j=1
d = d(x1, . . . xN, xN+1) =
N
j=1
x 2 j +
4
xm+2
(m + 2) 2 N+1
(4.8)
Proof: Elementary calculation shows that the defining system (4.3) for the
coefficients {X j } N+1
j=1 of an arbitrary conformal Killing field is, for xN+1 = 0,
equivalent to the system
2xN+1∂jX j = mX N+1 + 2
N + 1 xN+1divgX, j = 1, . . . N (4.9)
2xN+1∂N+1X N+1 = 2
N + 1 xN+1divgX (4.10)
x m N+1∂jX N+1 + ∂N+1X j = 0, j = 1, . . . N (4.11)
∂jX k + ∂kX j = 0, 1 ≤ j < k ≤ N (4.12)
Combining (4.9) and (4.10) and setting y = xN+1, X N+1 = η, and X i = ξ i
for 1 ≤ i ≤ N shows that each solution to (4.9) − (4.12) is a solution of

Conformal groups for operators of mixed and degenerate types 9
(2.18) − (2.20). For N ≥ 2, this latter system was shown to have a finite
dimensional solution space with the basis corresponding to (4.4) − (4.7).
Hence, any conformal Killing field gives rise to a one parameter symmetry
group for L.
To finish the proof, consider any regular solution of the full system (2.15)−
(2.20) describing the symmetry group, with a fixed (ξ 1 , . . . ξ N , η) equal to
(X 1 , . . . , X N , X N+1 ) satisfying (2.18) − (2.20). For this solution, the additional
constraints (2.15) − (2.17) force the stronger form (4.9) − (4.10) of
(2.18) to hold as well. Hence each generator of a one dimensional symmetry
group projects onto a conformal Killing field.
The explicit representation of the conformal algebras away from the metric
singularity allows for a trivial extension across the singularity by smoothness.
We formalize this observation in the following Theorem whose proof is
a direct consequence of Lemma 4.2.
Theorem 4.3. Let K be a pure power type change function of the form
(3.3) on RN+1 with N ≥ 2. Then the generators (4.4) − (4.7) of the conformal
algebras Kg(R N+1
± ) are all tangential to the singular hypersurface
Σ = {xN+1 = 0} in the sense that limxN+1→0 XN+1 = 0 and have a smooth
extension across Σ. The resulting Lie algebra Kg(RN+1 ) agrees with the natural
projection onto T RN+1 of the Lie algebra spanned by (2.10) − (2.13)
which generates the non-trivial part of the symmetry group for the operator
(1.1).
In this way, one could define the conformal group associated to the singular
metric g as the group of local diffeomorphisms generated by Kg(R N+1 )
which by Theorem 4.3 is seen to be isomorphic to the non-trivial part of
the symmetry group for the operator (1.1). We remark that the same result
holds for all of the cases (DE), (DH), and (MT) with any m > 0 where one
only needs to replace x m+2
N+1 in (4.8) with ±|xN+1| m+2 or xN+1|xN+1| m+1 respectively
and to accept limited smoothness in the coefficients of the Killing
fields across Σ.
In the planar case (N = 1), the situation is more complicated as one
might expect if one thinks of the limiting case m = 0. One expects a conformal
group of infinite dimension which turns out to be the case. However, the
symmetry group for the operator (2.1) has been shown to be finite dimensional;
hence there is only a one way correspondence between the conformal
transformations and the symmetries for (2.1). One recovers the correspondence
for the geometric operator (3.2). We begin with a description of the
conformal algebras away from the metric singularity.
Lemma 4.4. Let K be a pure power type change function of the form (3.3)
on a half-space R 1+1
± . Then the conformal algebra Kg(R 1+1
± ) is infinite dimensional
and each conformal Killing field is tangential to singular line
Σ = {x2 = 0} and extends smoothly across Σ.
Proof: The defining equations (4.3) for the coefficients of the conformal
Killing fields reduce to the pair of equations

10 K. R. Payne
x2∂1X 1 − x2∂2X 2 − m
2 X2 = 0 (4.13)
x m 2 ∂1X 2 + ∂2X 1 = 0 (4.14)
where we have used also divgX = ∂1X 1 + ∂2X 2 − mX 2 /(2x2). Hence the
system is equivalent to the corresponding (ξ, η) = (X 1 , X 2 ) part of the system
(2.18) and (2.20) describing the symmetry group. These two equations in two
unknowns have an infinite number of independent solutions including those
corresponding to the translation, dilation and inversion; namely
(X 1 , X 2 ) (1) = (1, 0), (4.15)
(X 1 , X 2 ) (2) = ((m + 2)x1, 2x2), (4.16)
(X 1 , X 2 ) (3) = ((m + 2) 2 x 2 1 − 4x m+2
2 , 4(m + 2)x1x2) (4.17)
as well as a remarkable sequence of independent solutions defined inductively
by (X 1 , X 2 ) (k) = (ξ (k), η (k)) with
ξ (k+1) = ξ 2 (k) − xm 2 η 2 (k) , and η (k+1) = 2ξ (k)η (k), k ≥ 2 (4.18)
as a simple calculation shows. Hence the conformal algebras have infinite
dimension.
All of the solutions above extend smoothly across the singular line x2 = 0
where the component X 2 tends to zero as x2 → 0; hence in this sense each X
is tangential to the singular line. Moreover, the equation (4.13) shows that
any C 1 continuation of X 2 must vanish as well.
As in the higher dimensional cases, we can now define the conformal algebra
Kg(R1+1 ) as the Lie algebra of smooth extensions across Σ of Kg(R 1+1
± )
and the conformal group associated to the singular metric g as the group of
local diffeomorphisms generated by Kg(R1+1 ) which is infinite dimensional.
On the other hand, the non-trivial part of the symmetry group associated to
the operator (1.1) is only three dimensional and corresponds to the generators
(4.15) − (4.17). The point being that the remainder of the system describing
the symmetry group (2.15) − (2.17) forces the pair (ξ, η) = (X1 , X2 ) to be
a linear combination of the generators (4.15) − (4.17) as has been shown in
[15]. On the other hand, if one considers the operator (3.2) one recovers the
complete conformal group as the non-trivial part of its symmetry group.
Theorem 4.5. Let K be a pure power type change function on R 1+1 . Then
the conformal algebra Kg(R 1+1 ) agrees with the natural projection onto T R 1+1
of the Lie algebra A(R 1+1 ) of generators of a one parameter symmetry group
for the operator
Lg = K(y)∂ 2 x + ∂ 2 y − m
2y ∂y. (4.19)
Proof: As a first step, we need to find the system analogous to (2.15) −
(2.20) which characterizes the generators v = ξ(x, y, u)∂/∂x+η(x, y, u)∂/∂y+
ϕ(x, y, u)∂/∂u of one parameter symmetry groups for the operator (4.19). In

Conformal groups for operators of mixed and degenerate types 11
this case, since the operator has a singular coefficient along Σ, we will use
the classification machinery away from Σ and then require smoothness in the
coefficients of v. Standard calculations for y = 0 show that: ξ = ξ(x, y), η =
η(x, y) are independent of u; ϕ = α(x, y)u + β(x, y) is linear in u; and
Lgα = Lgβ = 0 (4.20)
Lgξ = 2y m αx
Lgη + m
y ηy − m
η = 2αy
(4.22)
2y2 (4.21)
2y m ξx − 2y m ηy − my m−1 η = 0 (4.23)
y m ηx + ξy = 0 (4.24)
Relabelling by (x, y; ξ, η) = (x1, x2; X 1 , X 2 ) we see that (4.23)−(4.24) is precisely
the system (4.13) − (4.14). Hence each generator of a one dimensional
symmetry group must project onto a conformal Killing field. Conversely,
given a solution to (4.13) − (4.14), if one selects α = β = 0, one trivially
obtains (4.20) and verifies that (4.21)−(4.22) hold as well with αx = αy = 0.
Hence, each infinitesimal conformal transformation gives rise to an infinitesimal
symmetry, which completes the proof.
We close this section by noting that for the higher dimensional cases
N ≥ 2, one has the analogous result for the operator Lg and the same
kind of trivial splitting in the Lie algebra of generators A(R N+1 ). Also,
as one expects, the “isometry group” defined infinitesimally as the smooth
extensions of the Killing fields X (solutions of (4.1)) turn out to be generated
by the translations and rotations (for N ≥ 2) in the x-variables.
5 Concluding remarks
There are various uses of the term singular Riemannian/Lorenztian metric
in the literature. We have used the term in the sense of Hermann [12] in
which one has the presence of a smoothly varying symmetric bilinear but
degenerate form γ on the cotangent bundle T ∗ M of a smooth manifold M.
The particular nature of the singular geometry is described by the form
of the degeneracy in γ which in our situation involves a blow up in the
metric g at the boundary of open sets on which γ is non-degenerate. A
situation dual to the one we consider concerns not a blow up in the metric
tensor, but rather a degeneration in it. For example, a conical singularity
in the sense of Cheeger includes the relatively simple case of a metric of
the form g = dr 2 + r 2 ˜g on the cone C(N ) = R + × N where (N , ˜g) is an
n-dimensional Riemannian manifold. The Laplace-Beltrami operator in this
case is the (positive) operator
∆ = − ∂2 n
−
∂r2 r
∂
∂r
1
+ ∆˜g
(5.1)
r2

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

Conformal groups for operators of mixed and degenerate types 13
a sub-Lorentzian structure can be associated to our examples for which the
bicharacteristics project onto geodesics.
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