Page 58 - Curvature and Homology

40 I. RIEMANNIAN MANIFOLDS

1.1 1. Geodesic coordinates
In this section we digress to define a rather special system of local
coordinates at an arbitrary point Po of a Riemannian manifold M of
dimension n and metric g. But first, we have seen that the differential
equations of the auto-parallel curves uf = uuf(t), i = 1, ..., n of an affine
connection o$ = qk duk are given by

and that any integral curve of (1.11.1) is determined by a point Po
and a direction at Po. If the affine connection is the Levi Civita connection,
a geodesic curve (or, simply, geodesic) is defined as a solution of (1.1 1.1)
where the parameter t denotes arc length.
We define a local coordinate system (Ci) at Po as follows: At the pole Po
the partial derivatives of the components fij of the metric tensor vanish,
that is

Hence, the coefficients rf, of the canonical connection also vanish at Po:

Such a system of local coordinates is called a geodesic coordinatesystem.
Thus, at the pole of a geodesic coordinate system, covariant differentiation
is identical with ordinary differentiation. On the other hand, from
(1.11.1)

-a property enjoyed by the geodesics of En relative to a system of
cartesian coordinates. These are the reasons for exhibiting such coor-
dinates at a point of a Riemannian manifold. Indeed, in a given com-
putation substantial simplifications.may result.
The existence of geodesic coordinates is easily established. For, if we
write the equations of transformation (1.7.4) of an affine connection
in the form