3.2. ANALYSIS ON RIEMANNIAN MANIFOLDS 133
which is the formula (3.2.36).
The Gauss formula (3.2.36) can be generalized to more general gradient and divergent
operators, denoted byDAand divA.
LetAbe a vector field or a covector field, andu∈L^2 (TrkM). We define the operatorsDA
and divAby
(3.2.38)
DAu=Du+u⊗A,
divAu=divu−u·A.Based on (3.2.36), it is readily to verify that the following formula holds true for the operators
(3.2.38).
(3.2.39)
∫M(DAu,v)√
−gdx=−∫M(u,divAv)√
−gdx.Remark 3.12.The motivation to generalize the Gauss formulas (3.2.36) to the operatorsDA
and divAis to develop a new unified field theory for the fundamental interactions. The vector
fieldsAin (3.2.38) and (3.2.39) represent gauge fields in the interaction field equations, and
lead to a mass generation mechanism based on a first principle, called PID, different from the
famous Higgs mechanism.
3.2.5 Partial Differential Equations on Riemannian manifolds
To develop an orthogonal decomposition theory for general(k,r)-tensor fields, we need to
introduce the existence theorems for linear elliptic and hyperbolic equations on closed Rie-
mann and Minkowski manifolds. The existence results are well-known. In the following,
we give the definition of weak solutions and the basic existence theorems for PDEs without
proofs.
Linear elliptic equations
Consider the following PDEs defined on a Riemannian manifold{M,gij}with∂M=/0:(3.2.40) gijDiDju=divf+g,
whereDiare the covariant derivative operators, the unknown functionu:M→TrkMis a
(k,r)tensor field,g:M→TrkMandf:M→Trk+^1 M(orf:M→Trk+ 1 M)are given.
We need to introduce the concept of weak solutions for (3.2.40).
Definition 3.13.Let f∈L^2 (Trk+^1 M)(or f∈L^2 (Trk+ 1 M))and g∈L^2 (TrkM). A field u∈
H^1 (TrkM)is called a weak solution of (3.2.40), if for all v∈H^1 (TrkM)the following equality
holds true, ∫
M(∇u,∇v)√
−gdx=∫M[(f,∇v)−(g,v)]√
−gdx,where(·,·)is the inner product as defined in (3.2.33).
The following existence theorem is a classical result, which is a corollary of the well-
known Fredholm Alternative Theorem.