140 CHAPTER 3. MATHEMATICAL FOUNDATIONS
3.3.3 Uniqueness of orthogonal decompositions
In this subsection we only consider the case whereMis a closed manifold with zero first
Betti number.
In Theorem3.17, a tensor fieldu∈L^2 (TrkM)withk+r≥1 can be orthogonally decom-
posed into
(3.3.22)
u=∇φ+v for general closed manifolds,
u=∇φ+v+h for compact Riemannian manifolds.Now we address the uniqueness problem of the decomposition (3.3.22). In fact, ifuis a
vector field or a covector field:
u∈L^2 (TM) or u∈L^2 (T∗M),then the decomposition (3.3.22) is unique.
We see that ifu∈L^2 (TrkM)withk+r≥2, then there are different types of decompo-
sitions of (3.3.22). For example, foru∈L^2 (T 20 M), in a local coordinate system,uis given
by
u={uij(x)}.
In this case,uadmits two types of decompositions
(3.3.23) uij=Diφj+vij, Divij= 0 ,
(3.3.24) uij=Djψi+wij, Djwij= 0.
It is easy to see that ifuij 6 =uji, then (3.3.23) and (3.3.24) can be two different decomposition
ofuij. Namely
{vij} 6={wij}, (φ 1 ,···,φn) 6 = (ψ 1 ,···,ψn).
The reason is that the two differential equations generating the two decompositions (3.3.23)
and (3.3.24) as
(3.3.25) DiDiφj=Diuij and DiDiψj=Diuji
are different becauseDiuij 6 =Diuji.
However for a symmetric tensor fielduij=uji, as
Diuij=Diuji,the two equations in (3.3.25) are the same. By the uniqueness of solutions of (3.3.25), the
two solutionsφjandψjare the same:
φi=ψi for 1≤i≤n.Thus (3.3.23) and (3.3.24) can be expressed as
(3.3.26) uij=Diφj+vij, Divij= 0 ,
(3.3.27) uij=Djφi+wij, Djwij= 0.
From (3.3.26) and (3.3.27) we can deduce the following theorem.