Mathematical Principles of Theoretical Physics

3.2 Analysis on Riemannian Manifolds.

3.2 Analysis on Riemannian Manifolds

3.2.1 Sobolev spaces of tensor fields

The gravitational field equations are defined on a Riemannianmanifold. To study these equa-
tions, it is necessary to introduce various types of function spaces on manifolds, which pos-
sess different differentiability. In particular, we need to introduce the concept of weak deriva-
tives.
LetMbe ann-dimensional manifold with metric{gij}. The following functions are
defined onM.

1.Lpspaces. For a real numberp( 1 ≤p<∞), we denote

(3.2.1) Lp(M⊗pEN) =

{

u:M→M⊗pEN

∣

∣

∣

∫

M

|u|p

√

−gdx<∞

}

,

where

√

−gdxis the volume element, and|u|is the modulus ofu. For example, ifu:M→
TMis a vector field, then
|u|=|gijuiuj|^1 /^2.

The space (3.2.1) is endowed withLp-norm as

||u||Lp=

[∫

M

|u|p

√

−gdx

] 1 /p

.

By Functional Analysis, the spacesLp(M⊗pEN)are Banach spaces, and for 1<p<∞,
Lp(M⊗pEN)are reflective and separable. The dual spaces ofLp(M⊗pEN) ( 1 <p<∞)
are
Lp(M⊗pEN)∗=Lq(M⊗p(EN)∗),
Lq(M⊗pEN)∗=Lp(M⊗p(EN)∗),

1

p

+

1

q

= 1 ,

where(EN)∗is the dual space ofEN.
Forp=∞, we define that

L∞(M⊗pEN) ={u:M→M⊗pEN|uis bounded almost everywhere}.

The norm ofL∞(M⊗pEN)is defined by

||u||L∞=sup

M

|u|.

The spaceL∞(M⊗pEN)is a Banach space, but not reflective and separable. The space
L∞(M⊗pEN)is the dual space ofL^1 (M⊗pEN), i.e.

L^1 (M⊗pEN)∗=L∞(M⊗p(EN)∗).

2.Weakly differentiable functions. A fieldu∈Lp(M⊗pEN)is calledk-th order weakly
differentiable, if each componentujisk-th order weakly differentiable, i.e. for eachujthere
exists uniquely a functionφsuch that for allv∈C 0 ∞(M)we have

(3.2.2)

∫

M

φv

√

−gdx= (− 1 )k

∫

M

uj∂kvdx.