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N m is called an immersion if the rank of f is 11, the dimension of the domain M, at all points of M.

Manifold, 15 'J'he one gap in the preceding discussion is the definition of a metric for p2 , The missing metric can be supplied by an appeal to Problem 3-1, which will Iatcr be used quite often, and which the reader should peruse sometime before J't·acling Chapter 3. Roughly speaking, it shows that things like ll'2 , whidI ought to be manifolds, are. ) For the present, how('ver, we wi)) obtain our metric by a trick that simultaneously provides an imbedding of ll'2 in �4.

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