3.3. ORTHOGONAL DECOMPOSITION FOR TENSOR FIELDS 137
1) The tensor field u can be orthogonally decomposed into(3.3.9) u=∇Aφ+v with divAv= 0 ,whereφ∈H^1 (Trk−^1 M)orφ∈H^1 (Trk− 1 M).2) IfMis a compact Riemannian manifold, then u can be orthogonallydecomposed into(3.3.10) u=∇Aφ+v+h,whereφand v are as in (3.3.9), and h is a harmonic field, i.e.divAh= 0 , ∇Ah= 0.In particular, the subspace of all harmonic tensor fields in L^2 (TrkM)is of finite dimen-
sional:(3.3.11)
H(TrkM) ={h∈L^2 (TrkM)|∇Ah= 0 ,divAh= 0 },and
dimH(TrkM)<∞.Remark 3.18.The above orthogonal decomposition theorem implies thatL^2 (TrkM)can be
decomposed into
(3.3.12)
L^2 (TrkM) =G(TrkM)⊕L^2 D(TrkM) for general case,
L^2 (TrkM) =G(TrkM)⊕H(TrkM)⊕L^2 N(TrkM) forMcompact Riemainnian.HereHis as in (3.3.11), and
G(TrkM) ={v∈L^2 (TrkM)|v=∇Aφ,φ∈H^1 (Trk− 1 M)},
L^2 D(TrkM) ={v∈L^2 (TrkM)|divAv= 0 },
L^2 N(TrkM) ={v∈L^2 D(TrkM)|∇Av 6 = 0 }.They are orthogonal to each other:
L^2 D(TrkM)⊥G(TrkM), L^2 N(TrkM)⊥H(TrkM), G(TrkM)⊥H(TrkM).Remark 3.19.The orthogonal decomposition (3.3.12) ofL^2 (TrkM)implies that if a tensor
fieldu∈L^2 (TrkM)satisfies that
〈u,v〉L 2 =∫M(u,v)√
−gdx= 0 ∀divAv= 0 ,thenumust be a gradient field, i.e.
u=∇Aφ for someφ∈H^1 (Trk−^1 M)orH^1 (Trk− 1 M).Likewise, ifu∈L^2 (TrkM)satisfies that
〈u,v〉L 2 = 0 ∀v∈G(TrkM),thenu∈L^2 D(TrkM). It is the reason why we define a divA-free field by (3.3.8).