文档介绍：Page 291
5
Tensors
In the previous chapter we studied vector fields and functions on manifolds. In this chapter these objects are
generalized to tensor fields,which are sections of vector bundles built out of the tangent bundle. This study
is continued in the next chapter when we discuss differential forms,which are tensors with special symmetry
properties. One of the objectives of this chapter is to extend the pull-back and Lie derivative operations
from functions and vector fields to tensor fields.
Tensors on Linear Spaces
Preparatory to putting tensors on manifolds,we first study them on vector spaces. This subject is an
extension of linear algebra sometimes called “multilinear algebra.” Ultimately our constructions will be
done on each fiber of the tangent bundle,producing a new vector bundle.
k
As in Chapter 2, E, F,... denote Banach spaces and L (E1,...,Ek; F) denotes the vector space of con-
∗
tinuous k-multilinear maps of E1 ×···×Ek to F. The special case L(E, R) is denoted E ,the dual space
{ }
of is finite dimensional and e1,...,e n is an ordered basis of E,there is a unique ordered basis of
∗{ 1 n} j j j
E ,the dual basis e ,...,e ,such that e ,ei = δi where δi =1ifj = i and 0 otherwise. Furthermore,
for each v ∈ E,
n n
i i
v = e ,v ei and α= α, ei e ,
i=1 i=1
for each α∈ E∗,where , denotes the pairing between E and E∗. Employing the summation convention
whereby summation is implied when an index is repeated on upper and lower levels,these expressions e
i i
v = e ,v ei and α= α, ei e .
As in Supplement ,if E is infinite dimensional,by E∗ we will mean another Banach space weakly
paired to E; it need not be the full functional analytic dual of E. In particular, E∗∗ will always be chosen
to be E. With these conventions,tensors are defined as follows.
Definition. For a vector space E we put
r r+s ∗∗ R
Ts (E)=L (E ,...,E , E,...,E; )
292 5. Tensors
∗ r
(r copies of E and s copie