A comparison was made between the geodesic coordinates
and the transformed model along the X, Y and Z coordinates and differential coordinates [increment of x], [increment of y], [DELTA]z were obtained.
If now an extra non gravitational force acts on the gyroscope and as a result the gyroscope moves into a world line that is different from a geodesic, then we can not simply introduce local geodesic coordinates
at every point on of this world line which makes the equation of motion for the spin d[S.sup.n]/dt [not equal] 0.
Since the metrics [Mathematical Expression Omitted] are in geodesic coordinates and have their curvatures and their covariant derivatives uniformly bounded, it follows by Corollary 4.10 in the next section that by passing to a subsequence we can guarantee that for each [Alpha] (and indeed for all [Alpha] by diagonalization) the metrics [Mathematical Expression Omitted] converge uniformly with all their derivatives to a smooth metric [G.sup.[Alpha]] on [E.sup.[Alpha]] (or [Mathematical Expression Omitted] or [Mathematical Expression Omitted]) which is also in geodesic coordinates.
First we look at geodesic coordinates. Recall that a metric [g.sub.ij](x)d[x.sup.i]d[x.sup.j] defined in a ball around the origin is in geodesic coordinates if every line through the origin is a geodesic (parametrized proportional to arc length) and if [g.sub.ij] = [I.sub.ij] at the origin x = 0.
The metric [g.sub.ij] is in geodesic coordinates if and only if
which is one set of equations for geodesic coordinates. Suppose this is true.
(since [I.sub.jk] is also in geodesic coordinates).
Hence [G.sub.kl][x.sup.k] = [I.sub.kl][x.sup.k] in geodesic coordinates.
Then we get [Mathematical Expression Omitted] which shows we are in geodesic coordinates. This proves the lemma.
Our application will be to the tensor [I.sub.jk] which gives the Euclidean metric as a tensor in geodesic coordinates. We have