Mathematical Principles of Theoretical Physics

132 CHAPTER 3. MATHEMATICAL FOUNDATIONS

which can be generalized to tensor fields on Riemann manifolds.
LetMbe a Riemannian manifold with∂M=/0. Then, there is an inner product defined
onTMdefined by
(X,Y) =gijXiXj ∀X,Y∈TM.

Also, there is an inner product onT∗M:

(X∗,Y∗) =gijXi∗Yj∗ ∀X∗,Y∗∈T∗M.

The tensor bundleTrkMis

TrkM=T︸M⊗ ··· ⊗︷︷ TM︸

k

⊗︸T∗M⊗ ··· ⊗︷︷ T∗M︸

r

.

The inner products onTMandT∗Minduce a natural inner product onTrkM:

(3.2.33) (u,v) =gi 1 j 1 ···gikjkgl^1 s^1 ···glrsruil 11 ······ilrkvsj 11 ······sjrr ∀u,v∈TrkM.

Hence we can define the inner product〈·,·〉onL^2 (TrkM)by

(3.2.34) 〈u,v〉L 2 =

∫

M

(u,v)

√

−gdx ∀u,v∈L^2 (TrkM),

where(u,v)is as in (3.2.33).
Furthermore, we can also define an inner product on the spacesHm(TrkM)as follows

(3.2.35) 〈u,v〉Hm=

∫

M

[(− 1 )m(∆mu,v)+ (u,v)]

√

−gdx ∀u,v∈Hm(TrkM),

where∆=div·∇.
In the following, we give the Gauss formula onTrkM, which are crucial for the orthogonal
decomposition theory in the next section.

Theorem 3.11(Gauss Formula). For any u∈H^1 (TrkM)and v∈H^1 (Trk+^1 M)(or v∈
H^1 (Trk+ 1 M)),

(3.2.36)

∫

M

(∇u,v)

√

−gdx=−

∫

M

(u,divv)

√

−gdx.

Formula (3.2.36) is a corollary of the classical Gauss formula

(3.2.37)

∫

M

divw

√

−gdx=

∫

∂M

w·nds.

In fact, letwk= (u,vk), then by (2.3.28) we have

divw=Dk(u,vk) = (∇u,v)+ (u,divv).

If∂M=/0, then we derive from (3.2.37) that
∫

M

divw

√

−gdx=

∫

M

[(∇u,v)+ (u,divv)]

√

−gdx= 0 ,