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Seminaire Bourbaki, 30, 1987-1988 - Exp.686-699 by Bourbaki N.

By Bourbaki N.

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Xi P Then the difference in the coordinates of A and B is x'(B)-x'(A) just the commutator of the exponentiation operators. Returning to the Taylor series, we can write d 1 d2 0(E3), [eedldh e'd/@] = 1 + e - d+X- - E22 - +dX2 . r, m] Fig. 2 1. Geometric interpretation of the Lie bracket [ as the open part of an incomplete parallelogram whose other sides are equal parameter increments along integral curves of j7 and W. 15 When is a basis a coordinate basis? 12) justifies the picture we have given for it.

Two arbitrary vector fields do not commute: if = d/dh and w = d/dp, then where the last line follows from relabelling the summation indices in the final sum of the middle quantity. Therefore, the commutator + d d dhdp d d dpdh is a vector field whose components do not vanish in general. If d/dh and d/dp are two elements of a basis, then they will not be expressible as derivatives with respect to any coordinates. Such a basis is a noncoordinate basis. It is important to realize that this distinction between coordinate and noncoordinate bases is one which can be made only over some region of the manifold, not at a single point.

4) is called a natural coordinate system for TM. 12 identifies a particular vector at each point of M, and so the curve defines a vector field on M. e. one which is nowhere parallel to a fiber) is called a cross-section of TM. Clearly, it is not usually meaningful to ask for the 'length' of the curve, and so here we have an example of a manifold on which one usually would not bother to define a metric. A general fiber bundle consists of a base manifold, which in our case is the curve M, and one fiber attached t o each point of the base space.

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