Straightforward proof of closedness
The formula for the basic form 
is:
This expression should be understood as follows. Notice that 
is
an element of 
, i.e. an inhomogeneous differential
form on 
with values in polynomial functions on 
. We then evaluate the
function on 
; for example the polynomial 
evaluates as follows:
is
an element of , i.e. an inhomogeneous differential
form on with values in polynomial functions on . We then evaluate the
function on ; for example the polynomial evaluates as follows:
|
We have to prove that 
defined in Eq. (16) is horizonthal and closed.
The horizonthality follows immediately from the definition, because:
defined in Eq. (16) is horizonthal and closed.
The horizonthality follows immediately from the definition, because:
is essentially the projector on the horizonthal forms and - is automatically horizonthal
It remains to prove that is a closed form. We have:
is a closed form. We have:
where the following notations are assumed:
- the symbol
denotes the symmetrically ordered expression:
in the first line, the
acts on everything that is
(after symmetrization) to the right of it
in particular, it acts on
, giving
, which cancels with the second line