Immersions

On page 19, O’Neill comments that the proof of Lemma 33 is a mild generalization of the proof of proposition 28. I think (2) \iff (3) requires spelling out.

Let \phi : M^m \longrightarrow N^n and let \xi = (y^1,\ldots,y^n) be a co-ordinate system at \phi (p). Let \zeta = (x^1,\ldots,x^m) be a co-ordinate system at p. Then by (2) \frac{\partial (y^i \circ \phi)}{\partial (x^j)} has rank m. Thus by exercise 7 and by re-arranging the co-ordinates if necessary, (y^1 \circ \phi,\ldots,y^m \circ \phi) forms a co-ordinate system for M^m on a neighbourhood \cal{W} of p.

For the reverse, note that since (y^1 \circ \phi,\ldots,y^m \circ \phi) is a co-ordinate system, by exercise 7, \frac{\partial (y^i \circ \phi)}{\partial x^j} has rank m.

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