Mathematical Principles of Theoretical Physics

150 CHAPTER 3. MATHEMATICAL FOUNDATIONS

3.4.4 Variational principle withdivA-free constraint

LetMbe a closed manifold. A Riemannian metricGonMis a mapping

G:M→T 20 M=T∗M⊗T∗M,

which is symmetric and nondegenerate. Namely, in a local coordinate system,Gcan be

expressed as

(3.4.27) G={gij} with gij=gji,

and the matrix(gij)is invertible onM:

G−^1 = (gij) = (gij)−^1 :M→T 02 M=TM⊗TM.

If we regard a Riemannian metricG={gij}as a tensor field on the manifoldM, then

the set of all metricsG={gij}onMconstitute a topological space, called the space of

Riemannian metrics onM. The space of Riemannian metrics onMis defined by

Wm,^2 (M,g) ={G|G∈Wm,^2 (T 20 M),G−^1 ∈Wm,^2 (T 02 M),

Gis the Riemannian metric onMas in( 3. 4. 27 )}.

The spaceWm,^2 (M,g)is a metric space, but not a Banach space. However, it is a subspace

of the direct sum of two Sobolev spacesWm,^2 (T 20 M)andWm,^2 (T 02 M):

Wm,^2 (M,g)⊂Wm,^2 (T 20 M)⊕Wm,^2 (T 02 M).

A functional defined onWm,^2 (M,g):

(3.4.28) F:Wm,^2 (M,g)→R

is called the functional of Riemannian metric. In general, the functional (3.4.28) can be

expressed as

(3.4.29) F(gij) =

∫

M

f(gij,···,∂mgij)

√

−gdx.

Since(gij)is the inverse of(gij), we have

(3.4.30) gij=

1

g

×the cofactor ofgij.

Therefore,F(gij)in (3.4.29) also depends ongij, i.e. putting (3.4.30) in (3.4.29) we get

(3.4.31) F(gij) =

∫

M

f ̃(gij,···,∂mgij)√−gdx.

We note that althoughWm,^2 (M,g)is not a linear space, but for a given elementgij∈
Wm,^2 (M,g)and any symmetric tensor fieldsXij,Xij, there is a numberλ 0 >0 such that

(3.4.32)

gij+λXij∈Wm,^2 (M,g) ∀ 0 ≤ |λ|<λ 0 ,

gij+λXij∈Wm,^2 (M,g) ∀ 0 ≤ |λ|<λ 0.