3.1. BASIC CONCEPTS 113
the inner product ofXandYis defined by
(3.1.18) 〈X,Y〉=gij(p)XiYj.
It is clear that the inner product (3.1.18) is an invariant. The angle betweenXandYis defined
as
cosθ=〈X,Y〉
|X||Y|
, |Z|=
√
gijZiZj forZ=X,Y.- The inverse of(gij), denoted by
(gij) = (gij)−^1 ,is a second order contra-variant tensor. In fact, by (3.1.16) we have that
I= ( ̃gij)( ̃gij) =(
∂ φ
∂x)− 1
(gij)[(
∂ φ
∂x)T]− 1
(g ̃ij),whereIis unit matrix. It follows that
(g ̃ij) =(
∂ φ
∂x)T
(gij)(
∂ φ
∂x)
.
Hence{gij}is a second-order contra-variant tensor.
3.1.2 Physical fields and vector bundles
LetMbe a manifold. A vector bundle onMis obtained by gluing anN-dimensional linear
spaceENat each pointp∈M, denoted by
(3.1.19) M⊗pEN
def
⋃p∈M{p} ×ENp,whereEN=RN, orCN, or the Minkowski space.
In a vector bundle (3.1.19), the geometric position ofENp is related withp∈M. For
example, the set of all tangent spaces is a vector bundle onM, called the tangent bundle of
M, denoted by
TM=M⊗pTpM.
For (3.1.19), if the positions of bundle spacesENp(p∈M)are independent ofp, then it
is called a geometrically trivial vector bundle. Vector bundles on a flat manifold are always
geometrically trivial.
The physical background of vector bundles are very clear. Each physical field must be
a mapping from the base spaceMto a vector bundleM⊗pEN. Actually, all physical
events occur in the space-time universeM, and the physical fields describing these events
are defined on vector bundles. This point of view is well illustrated in the following examples.