Singular metrics and associated conformal groups underlying ...

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)