It’s now time to do some more manipulations with differential forms on a Riemann surface. This will establish several notions we will need in the future.
The Hodge star
Given the 1-form in local coordinates as
, define
In other words, given the decomposition , we act by
on the first sumamand and by
on the second. This shows that the operation is well-defined. Note that
is conjugate-linear and
. Also, we see that
if
. This operation is called the Hodge star.
From the latter description of the Hodge star we see that for any smooth ,
From the definitions of , this can be written as
if
is the usual Laplacian with respect to the local coordinates
.
The Hodge star allows us to define co things. A form is co-closed if
; it is co-exact if
for
smooth. (more…)