3.3 Orthogonal Decomposition for Tensor Fields
Remark 3.16.IfM=R^1 ×M ̃with metric (3.2.43) and∂M ̃=/0, then the problem (3.2.45)-
(3.2.46) is replaced by the following initial value problem:
(3.2.47)
−
∂^2 u
∂t^2+gijDiDju=divf,u( 0 ) =φ,
ut( 0 ) =ψ.The same existence theorem holds true as well for problem (3.2.47).
3.3 Orthogonal Decomposition for Tensor Fields
3.3.1 Introduction
LetMbe a Riemannian manifold (or a Minkowski manifold),∂M=/0, anduis a tensor
field onM:
(3.3.1) u:M→TrkM.
The orthogonal decomposition of tensor fields means that thefieldugiven by (3.3.1) can be
decomposed as
(3.3.2) u=∇φ+v and divv= 0 ,
for someφ:M→Trk−^1 M(orφ:M→Trk− 1 M). Moreover∇φandvare orthogonal in the
following sense:
(3.3.3) 〈∇φ,v〉=
∫M(∇φ,v)√
−gdx= 0.In this section, we shall show that all tensor fields as given by (3.3.1) can be decomposed
into the form (3.3.2) satisfying (3.3.3).
In order to understand the problem well, we first introduce some classical results: the
Helmholtz decomposition and the Leray decomposition onM=Rn.
1.Helmholtz decomposition.Letu∈L^2 (TR^3 )be a 3-dimensional vector field, i.e.u(x) = (u^1 (x),u^2 (x),u^3 (x)) forx∈R^3 ,then there exist a functionφ∈H^1 (R^3 )and a vector fieldA∈H^1 (TR^3 ), such thatucan be
decomposed as
(3.3.4)
u=∇φ+curlA,
∫R^3∇φ·curlAdx= 0.Note that div(curlA) =0. Hence, the Helmholtz decomposition is an important initial result
on orthogonal decompositions.