ABSTRACT
The initial conditions corresponding to the fields Y/ on S' lead to a unique solution and thus the fields}j are linear combinations of fields V; . Therefore, we can conclude because we have constructed m fields X m and Y, forming a coordinate basis for an m-dimensional submanifold. Note we must emphasize the "Lie transport law" safeguards the commutative property since:
[Xm,[Y"YJ]] +[Y"[Yl'Xm ]]+[YJ,[Xm,Y,]] =0 where the last two terms are zero, and thus
[Y"Yj ] = o. The proof is so established.
