Singular metrics and associated conformal groups underlying ...

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