Page 27 - Curvature and Homology

1.3. TENSOR BUNDLES 9
elements of V and those of type (0,l) with elements of V* by taking
into account the duality between V and V*. Hence Ti V and V*.
The tensor space T,1 may be considered as the vector space of all
multilinear maps of VX ... x V (r times) into V. In fact, given f E Tt,
a multilinear map t: V x . . . x V -+ V is uniquely determined by the
relation
(t(vl, ..., vr),v*) = f(vl, ..., vr,v*) E R (1 -2.15)

for all v,, ..., v, E V and v* E V*, where, as before, ( , ) denotes the
value which v* takes on t(vl, ..., v,). Clearly, this establishes a canonical-
isomorphism of T,1 with the linear space of all multilinear maps of
Vx ...x V into V. In particular, Ti may be identified with the space
of all linear endomorphisms of V.
Let (e,) and (e*k) be dual bases in V and V*, respectively:

These bases give rise to a base in Ti whose elements we write as
kl...k, = eil @ ... @ eir @ e*k1 @ ... @ e*ke (cf. I. A for a defini-
ef l...ir
tion of the tensor product). A tensor t G Ti may then be represented in
the form
t = @..Ar kl...k8,
kl,..k,%l...ir (1.2.17)
that is, as a linear combination of the basis elements of T,'. The coefficients
&..*. kl...k, then define t in relation to the bases {ei} and {c*~).

1.3. Tensor bundles
In differential geometry one is not interested in tensors but rather .
in tensor fields which we now proceed to define. The definition given
is but one consequence of a general theory (cf. I. J) having other
applications to differential geometry which will be considered in 5 1.4
and 5 1.7. Let Ti(P) denote the tensor space of tensors of type (r, s)
over Tp and put
9'-i = (J TI(P).
PEM

We wish to show that 9; actually defines a differentiable manifold and
that a 'tensor field' of type (r, s) is a certain map from M into Ff, that
is a rule which assigns to every P E M a tensor of type (r, s) on the
tangent space Tp. Let V be a vector space of dimension n over R and T: