Abstract
This is a footnote of a recent interesting work of Cohen, Manin and Zagier, where they, among other things, produce a natural isomorphism between the sheaf of (n-1)-th order jets of the n-th tensor power of the tangent bundle of a Riemann surface equipped with a projective structure and the sheaf of differential operators of order n (on the trivial bundle) with vanishing 0-th order part. We give a different proof of this result without using the coordinates, and following the idea of this proof we prove:
Take a line bundle L with
L
2
=T
on a Riemann surface equipped with a projective structure. Then the jet bundle
J
n
(
L
n
)
has a natural flat connection with
J
n
(
L
n
)=
S
n
(
J
1
(L))
. For any
m>n
the obvious surjection
J
m
(
L
n
)→
J
n
(
L
n
)
has a canonical splitting. In particular, taking
m=n+1
, one gets a natural differential operator of order
n+1
from
L
n
to
L
−n−2
.