Mathematical Principles of Theoretical Physics

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.