Gravity and Strings

24 Differential geometry
Since ⋆d ⋆
ω ρ1···ρk−1 = (−1)k(d−k+1)−1 sign g ∇µω µ ρ1···ρk−1 , (1.133)
we find that the relation between δ and the divergence is
(δω)ρ1···ρk−1 = (−1)d∇µω µ ρ1···ρk−1 . (1.134)
Only k-forms can be integrated on k-dimensional manifolds. If ω is a (d − 1)-form defined
on a d-dimensional manifold M with boundary ∂M, then Stokes’ theorem states that
dω = ω. (1.135)
M
It is convenient to define volume forms for a manifold and its lower-dimensional submanifolds.
Their contraction with other tensors results in differential forms that can be
integrated. Thus, we define in a d-dimensional manifold, for (d − n)-dimensional submanifolds
M d−n ,0≤ n ≤ d, the volume forms
∂M
d d−n µ1···µn ≡ dxν1 1
νd−n ···dx
(d − n)! √ ɛν1···νd−nµ1···µn . (1.136)
|g|
Observe that the standard invariant-volume form for the total manifold M d is just d d
up to a sign (we now use the signature (+−···−)):
d d = (−1) d−1 dx 1 ∧ ···∧ dx d |g|≡(−1) d−1 d d x |g|. (1.137)
Now, if we have a rank-n completely antisymmetric contravariant tensor T µ1···µn and
contract it with the volume element dd−nµ1···µn ,wehave constructed a (d − n)-form that
can be integrated over a (d − n)-dimensional submanifold. Up to numerical factors, that
form is the Hodge dual of the n-form that one gets by lowering the indices of Tµ1···µn :
1
n! dd−n µ1···µn T µ1···µn = ⋆ T. (1.138)
We can also take the divergence of the tensor and contract it with the volume element
dd−n−1µ1···µn−1 . The result is
(−1) d−n
(n − 1)! dd−n+1 µ1···µn−1 ∇ρT ρµ1···µn−1 = d ⋆ T. (1.139)
Stokes’ theorem for the exterior derivative of form ⋆T integrated over a (d − n + 1)dimensional
submanifold Md−n+1 with (d − n)-dimensional boundary ∂Md−n+1 is now
dd−n+1µ1···µn−1∇ρT ρµ1···µn−1 = (−1)d−n
d
n
d−nµ1···µn T µ1···µn . (1.140)
M d−n+1
The n = 1 case is the Gauss–Ostrogradski theorem,
d d x |g|∇µv µ = (−1) d−1
M d
∂M
∂M d−n+1
d d−1 µv µ . (1.141)