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O t This is Df('Y(O)) · 'Y'(O) by the chain rule, unsurprisingly since 'Y(O) = p and 'Y'(O) = v.

So, denoting by {(x, x)} CM x M, we consider the map g: M X M\~ - sn-1;(x, y) 33 ~ the set y-x f--+ - - - lly-xll" The domain of g has dimension 2m so again, provided 2m < n - 1, the image has measure zero. 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.

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An Introduction to Differential Manifolds by Dennis Barden

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