Page 46 - Curvature and Homology

28 I. RIEMANNIAN MANIFOLDS

Clearly, the fz,, i, k = 1, ..., n are the elements of a non-singular
matrix (2yd ,). Conversely, every non-singular matrix defines a frame
expressed in the above form. The set of all frames at all points of M
can be giv'en a topology, and in fact, a differentiable structure by taking
d, ..., un and (fFi ,) as local coordinates in n-l(U). The differentiable
manifold B is called the bundle of frames' or bases over M with structural
p p qn, R).
Let (f:)) denote the inverse matrix of (f:,,). In the overlap of two
coordinate neighborhoods, (ud, f:,) and (Cd, [:,) are related by

It follows that

Hence, for each i, the function 41tf) assigns to every point x of n-l(U)
a 1-form ad = f7)duj at ~(x) in U. Defining 8C = n*ad, i = 1, ..., n
we obtain n linearly independent l-forms 8C on the whole of B. Now,
we take the covariant differential of each of the vectors Xd. From (1.7.7)
and (1.7.8) we obtain

and so from (1.8.3)

Denoting the common expression in (1.8.6) by or," we see that the or,"
define n2 linear differential forms 8 = w*at on the whole of the
bundle B.
The n2 + n forms 8C, 8: in B are vector-valued differential forms in B.
To see this, identify B with the collection of vector space isomorphisms
x : IF-* Tp; namely, if x is the frame {XI, ..., Xn) at P, then x(al, ..., an)
= aCXe Now, for each t E T,, define 8 to be an Rn-valued l-form by