3.4. VARIATIONS WITHDIVA-FREE CONSTRAINTS 151
With (3.4.32), we can define the following derivative operators of the functionalF:δ∗F:Wm,^2 (M,g)→W−m,^2 (T 02 M),
δ∗F:Wm,^2 (M,g)→W−m,^2 (T 20 M),whereW−m,^2 (E)is the dual space ofWm,^2 (E), andδ∗F,δ∗Fare defined by
〈δ∗F(gij),X〉=d
dλ∣
∣
∣
λ= 0(3.4.33) F(gij+λXij),
〈δ∗F(gij),X〉=d
dλ∣
∣
∣
λ= 0(3.4.34) F(gij+λXij).
For any give metricgij∈Wm,^2 (M,g), the value ofδ∗Fandδ∗Fatgijare second-order
contra-variant and covariant tensor fields:
(3.4.35)
δ∗F(gij):M→TM×TM,
δ∗F(gij):M→T∗M×T∗M.Theorem 3.24.Let F be the functionals defined by (3.4.29) and (3.4.31). Then the following
assertions hold true:
1) For any gij∈Wm,^2 (M,g),δ∗F(gij)andδ∗F(gij)are symmetric tensor fields.2) If{gij} ∈Wm,^2 (M,g)is an extremum point of F, i.e.δF(gij) = 0 , then{gij}is also
an extremum point of F.3) δ∗f andδ∗F have the following relation(δ∗F(gij))kl=−gkrgls(δ∗F(gij))rs,where(δ∗F)kland(δ∗F)klare the components ofδ∗F andδ∗F.Proof.We only need to verify Assertion (3). In view ofgikgk j=δij,we have the variational
relation
δ(gikgk j) =gikδgk j+gk jδgik= 0.
It implies that
(3.4.36) δgkl=−gkigl jδgij.
In addition, in (3.4.33) and (3.4.34),
λXij=δgij, λXij=δgij, λ 6 =0 small.Therefore, by (3.4.36) we get
〈(δ∗F)kl,δgkl〉=−〈(δ∗F)kl,gkigl jδgij〉=−〈gkigl j(δ∗F)kl,δgij〉=〈(δ∗F)ij,δgij〉.Hence we have
(δ∗F)ij=−gkigl j(δ∗F)kl.
Thus Assertion (3) follows and the proof is complete.