Page 60 - Curvature and Homology

42 I. RIEMANNIAN MANIFOLDS

3. If P is a vector space over F, zi : V x W-+ P is a bilinear map onto P, and
if for any vector space Z,8 : V x W-h Z (8, bilinear), there is a unique linear
rnapB:p-+Z with B.li=e,
li
vxw- P

then P and V @ W are canonically isomorphic.

We are now able to give an important alternate construction of the tensor
product. The importance of this construction rests in the fact that it is a typical
example of a more general process, viz., dividing free algebras by relations.
the
be
4. Let Fvx , free vector space generated by V x Wand consider V x W
as a subset of Fvx with the obvious imbedding. Let K be the subspace of
FYx generated by elements of the form

Then, (FV, ,)/K together with the projection map u : V x W+ (FVx ,)/K
satisfies the characteristic property for the tensor product of V and W. In
particular, u is bilinear. It follows that (FV, ,)/K is canonically isomorphic
with V @ W.
In the following exercise we discuss the concept of a tensorial form.
5. By a tensmal p-form of type (r, s) at a point P of a differentiable manifold
M we shall mean an element of the tensor product of the vector space TJ(P)
of tensors of type (r, s) at P with the vector space AP(Tp) of p-forms at P.
A tensorial p-form of type (r, s) is a map M -+ T,' @ Ap(T) assigning to each
P E M an element of the tensor space T,'(P) @ AqTp). A tensorial p-form of
type (0,O) is simply a p-form and a tensorial 1-form of type (1,O) or vectorial
l-form may be considered as a 1-form with values in T.
Show that a tensorial p-form of type (r,s) may be expressed as a p-form
whose coefficients are tensors of type (r, s) or as a tensor field of type (r;s)
with p-forms as coefficients. -
6. The notation of the latter part of 5 1.9 is employed in this exercise. We shall
use the symbol P' to denote the position vector OP' relative to some fixed
point 0 E An. Then, the vectors e; may be expressed as