3.1. BASIC CONCEPTS 117
As discussed in Subsection2.1.5, each symmetry possesses three ingredients: space
(manifold), transformation group, and tensors. Consequently, the spacesMare different
for different symmetries. However, in the fashion providedby (3.1.31)-(3.1.33), the base
spaceMis fixed, and all symmetric transformations as groups act on the bundle spacesEN
as in (3.1.32), and the fields of (3.1.31) are automatically transformed as in (3.1.33). Thus,
the physical invariances are referred to a few group actingson bundle spaces, i.e. the linear
transformation onEN.
To illustrate this idea, we start with general tensors on ann-dimensional Riemannian man-
ifoldM. Let two coordinate systemsx= (x^1 ,···,xn)andx ̃= (x ̃^1 ,···,x ̃n)are transforming
under the following coordinate transformation:
(3.1.34) ̃xk=φk(x), 1 ≤k≤n.
Then(k,r)-tensors transform under
(3.1.35) T ̃ji 11 ······jikr=blj^11 ···blrjrais^11 ···aiskkTl 1 s^1 ······lsrk,
whereaij=∂ ̃xi/∂xj,bij=∂xi/∂ ̃xj.
Now, we consider the general tensors from the viewpoint of vector bundles. A(k,r)
tensor field is a mapping:
(3.1.36) T:M→TrkM=M⊗pEp,
and the bundle spaceEpis given by
(3.1.37) Ep=TpM⊗ ··· ⊗TpM
︸ ︷︷ ︸
k
⊗Tp∗M⊗ ··· ⊗Tp∗M
︸ ︷︷ ︸
r.
In thex-coordinate system, the bases ofTpMandTp∗Mare:
(3.1.38)
TpM:ei=∂/∂xi for 1≤i≤n,
Tp∗M:ei=dxi for 1≤i≤n.These bases induce a basis of the linear spaceEpin (3.1.37):
eij 11 ······ikjr=ei 1 ⊗ ··· ⊗eik⊗ej^1 ⊗ ··· ⊗ejr,and the field (3.1.36) can be expressed as
(3.1.39) T=Tji 11 ······ijkreij 11 ······ikjr.
When we take linear transformations onTpMandTp∗M:
(3.1.40)
A:TpM→TpM, A= (aij),
B:Tp∗M→Tp∗M, B= (bij) = (AT)−^1 ,then the fieldTof (3.1.36) (i.e. (3.1.39)) will transform in the same fashion as (3.1.35).