3.4. VARIATIONS WITHDIVA-FREE CONSTRAINTS 147
For the general form of Yang-Mills functional given by(3.4.10) F=
∫M[
−
1
4
GabFμ νaFμ νb]
dx,where(Gab)is the Riemann metric onSU(N)given by (2.4.49). The derivative operator ofF
in (3.4.10) is as follows
(3.4.11) δF=Gab∂αFα βb −ggα μGbcλdacFα βbGdμ.
3.4.3 Derivative operator of the Einstein-Hilbert functional
The Einstein-Hilbert functional is in the form
(3.4.12) F=
∫MR
√
−gdx,whereMis ann-dimensional Riemannian manifold with metricgij, andR=gijRijis the
scalar curvature ofM, andRijis the Ricci curvature tensor:
Rij=1
2
gkl(
∂^2 gkl
∂xi∂xj+
∂^2 gij
∂xk∂xl−
∂^2 gik
∂xj∂xl−
∂^2 gjl
∂xi∂xk)
(3.4.13) +gklgrs(ΓrklΓijs−ΓrikΓsjl),
and the Levi-Civita connectionsΓrkl are written as
(3.4.14) Γrkl=
1
2
grs(
∂gks
∂xl+
∂gls
∂xk−
∂gkl
∂xs)
.
First we verify the following derivative operatorδFof the Einstein-Hilbert functional
(3.4.12)-(3.4.14):
(3.4.15) δF=Rij−
1
2
gijR.Note thatgijandgijhave the relationsgij=1
g(3.4.16) × ||gij||, ||gij||the cofactor ofgij,
gij=1
g(3.4.17) × ||gij||, ||gij||the cofactor ofgij.
Hence by (3.4.16), we have
d
dλ∣
∣
∣
λ= 0(3.4.18) det(gij+λ ̃gij) = ̃gij× ||gij||= ̃gijgijg.
In addition, bygikgk j=δij, we obtain
d
dλ∣
∣
∣
λ= 0(gik+λg ̃ik)(gk j+λg ̃k j) = 0.