3.4. VARIATIONS WITHDIVA-FREE CONSTRAINTS 155
If the first Betti numberβ 1 (M) 6 =0, then we have the following theorem.
Theorem 3.29.Let the first Betti number ofMisβ 1 (M) =N with N 6 = 0. Then there are
a scalar fieldφ∈H^2 (M)and N vector fieldsψj( 1 ≤j≤N)in H^1 (T∗M)such that the
extremum point{gij}of F with divergence-free constraint satisfies the equation
(δF(gij))kl=DkDlφ+
N
∑
j= 1
(3.4.50) αjDkψlj,
(3.4.51) DkDkψlj=−Rklψkj for 1 ≤j≤N, 1 ≤l≤n,
whereαj( 1 ≤j≤N)are constants, Rkl=gk jRjland Rjlare the Ricci curvature tensors.
Proof.Since the first Betti numberβ 1 (M) =N( 6 = 0 ), by the de Rham theorem, there areN
closed 1-forms
(3.4.52) ωj=ψkjdxk∈Hd^1 (M) for 1≤j≤N,
they constitute a basis for the 1-dimensional de Rham homologyHd^1 (M). Henceωj( 1 ≤
j≤N)are not exact, and satisfy that
dωj=
(
∂ ψkj
∂xl
−
∂ ψlj
∂xk
)
dxl∧dxk= 0 ,
which imply that
∂ ψkj
∂xl
=
∂ ψlj
∂xk
for 1≤j≤N.
or equivalently
Dlψkj=Dkψlj for 1≤j≤N.
Namely,∇ψj∈L^2 (T∗M⊗T∗M)are symmetric second-ordercovariant tensor fields. Hence
any covector fieldΦ∈L^2 (T∗M)satisfying
(3.4.53) DlΦk=DkΦl
must be in the following form
(3.4.54) Φk=Dkφ+
N
∑
j= 1
αjψkj,
whereαj( 1 ≤j≤N)are constants,φis some scalar field. Hence, byA=0 inDA, the
equation (3.4.42) in Theorem3.26can be expressed as
(δF(gij))lk=DlΦk,
whereΦksatisfy (3.4.53) and (3.4.54), which are the equations given by (3.4.50).
On the other hand, by the Hodge decomposition theorem, the 1-forms in (3.4.52) are
harmonic, i.e.
dωj= 0 , δ ωj= 0 for 1≤j≤N.