Mathematical Principles of Theoretical Physics

3.2. ANALYSIS ON RIEMANNIAN MANIFOLDS 129

3) the Laplace operator:DkDk=div·∇,

4) the Laplace-Beltrami operator:∆=dδ+δd,

5) the wave operator:=DμDμ, withDμbeing the 4-D gradient operator.

We now give a detailed account on the above operators.

1.Gradient operator.The gradient operator∇is a mapping of the following types:

(3.2.20)

∇k:W^1 ,p(TrkM)→Lp(Trk+^1 M),

∇k:W^1 ,p(TrkM)→Lp(Trk+ 1 M),

and∇kand∇khave the relation

∇k=gkl∇l, ∇k=gkl∇l,

and{gkl}the Riemann metric ofM.∇is expressed as

(3.2.21)

∇k= (D 1 ,···,Dn),

Djthe covariant derivative operators.

2.Divergence operator.The divergence operator div is a mapping of the following types:

(3.2.22)

div :W^1 ,p(Trk+^1 M)→Lp(TrkM),

div :W^1 ,p(Trk+ 1 M)→Lp(TrkM).

ForT={Tji 11 ······ijkr+^1 } ∈W^1 ,p(Trk+^1 M)andT={Tji 11 ······ijrk+ 1 } ∈W^1 ,p(Trk+ 1 M),

(3.2.23)

divT=DilTji 11 ······ijlr···ik+^1 , and

divT=DjlTji 11 ······ijkl···jk+ 1

Asu∈W^1 ,p(TM)andu∈W^1 ,p(T∗M), we can give the expressions of divuin the
following.
Letu∈W^1 ,p(TM),u= (u^1 ,···,un). Then by (3.2.23),

divu=Dkuk=

∂uk

∂xk

+Γkk juj.

By the Levi-Civita connections (2.3.25), the contraction

Γkk j=

1

2

gkl

∂gkl

∂xj

=

1

2 g

∂g

∂xj

=

1

√

−g

∂

√

−g

∂xj

,

hereg=det(gij). Thus we have

(3.2.24) divu=
∂uk
∂xk

+

1

√

−g

∂

√

−g

∂xk

uk=

1

√

−g

∂(

√

−guk)

∂xk

.