- #1

- 55

- 9

where the referred mapping (12.1) is a map ##\varphi:I_k\to U_S(x_0)##, in which:

1. ##I_k\subset\mathbb{R}^n## is the k-dimensional unit cube,

2. ##x_0## is a generic point on the surface ##S## and ##U_S(x_0)## is a neighborhood of ##x_0## of points of ##S##.

I think that there is a deep significance in this theorem that I can't see yet, but anyway, looking brutally at the proof:

we can see that up to this point the author uses the inverse function theorem to say that, if we consider only the first k coordinates both in the domain and in the range, ## \varphi ## is actually a diffeomorphism between two suitable neighborhoods ## U_{ \mathbb {R}^k_t} (0) ## e ## U _ {\mathbb {R} ^ k_x} (\varphi (0)) ##.

Then, the proof continues in this way:

and here I don't understand why the whole ## f: U _ {\mathbb {R} ^ n_x} (\varphi (0)) \to U_ {\mathbb {R} ^ n_t} (0) ## (##n## instead of ##k##) should be a diffeomorphism. Indeed, it seems to me that ## t ^ {k + 1}, ..., t ^ n ## are always zero, as ##x## varies.