B' be coordinate bundles having the same fibre F and group G.

However, since the domain off is compact, its image is also compact and so closed. Thus the complement of f(a(M)) in sn-l is a nonempty open set which must therefore contain points v not in the image of g. As already explained, for such v, 1fv will be an embedding and this argument • can be repeated to obtain the required embedding of M in JR. 2 m+ 1 . 5. Vector Fields and I-forms on Euclidean Spaces. Although vector fields and I-forms are global concepts, when performing explicit calculations we shall almost always work locally.

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