Page 49 - Curvature and Homology

is independent of the choice of local coordinates ui. In this way, if we
are given a parametrized curve C(t), the integral

where X(t) is the tangent vector to C(t) defines the length s of the arc
joining the points (ui(t,,)) and (uyt,)).
Now, every differentiable manifold M (of class k) possesses a Rieman-
nian metric. Indeed, we take an open covering (Ua} of M by coordinate
neighborhoods and a partition of unity (gal subordinated to U,. Let
&:(= Ern_, dui dui) be a positive definite quadratic differential form
defined in each U, and let the carrier of g, be contained in U,. Then,
Zagah: defines a Riemannian metric on M.
Since the dui dui have coefficients of class k - 1 in any other coor-
dinate system and the g, can be taken to be of class k the manifold M
possesses a Riemannian metric of class k - 1.
It is now shown that there exists an affine connection on a differentiable
manifold. In fact, we prove that there is a unique connection with ihe
properties: (a) the twsion tensor is zero and (b) the scalar product (relative
to some metric) is preserved during parallel displacement. To show this,
assume that we have a connection I'jk satisfying conditions (a) and (b).
We will obtain a formula for the coefficients r;i in terms of the metric
tensor g of (b). Let X(t) = g?(t)(a/aui) and Y(t) = qi(t)(a/a~i) be
tangent vectors at the point (ui(t)) on the parametrized curve C(t). The
condition that these vectors be parallel along C(t) are

and

By condition (b),
d
--,(fir 13 =Os

Since (1.9.6) holds for any pair of vectors X and Y and any parametrized
curve C(t),
%
- .!?u Gk + gi, rh (1.9.8)
=