Blaga P. Lectures on the differential geometry of - tiera.ru

4.7. Differentiable maps on a surface 133
is a local parameterization of S around a such that the local representation fr = f ◦ r
is smooth. Obviously, it is enough to show that then the local representation of f in
any other parameterization of S around a is, also, smooth. We choose, thus, another
parameterization, (U1, r1), around a and let W = r(U) ◦ r1(U1). Then in r−1 (W) ⊂ U1,
can be represented as
fr1
fr1 = f ◦ r1 = f ◦ (r ◦ r −1 ) ◦ r1 = ( f ◦ r) ◦ (r −1 ◦ r1) = fr ◦ (r −1 ◦ r1).
Since both fr (from the hypothesis) and r−1 ◦ r1 (from the theorem 4.3.1) are smooth, it
follows that fr1 is, equally smooth. �
Example. Let S be a surface and (U, r) a local parameterization of S . As we explained
earlier, the map r −1 : r(U) → R 2 is nor smooth in the classical sense. The reason is that
its domain is not an open set of an Euclidean space, so the notion itself doesn’t make
sense for it. We also showed that, however, locally, r −1 is the restriction of a smooth map
defined on an open set of R 3 . The notion we just defined is, actually, the natural frame of
discussing this important map, which, in fact, assigns to each point of the surface (lying
in r(U), of course), a pair of coordinates. Indeed, as a map defined on an open set of
S , r −1 is smooth, as we can see easily, because the local representation of r −1 in the
parameterization (U, r) is � r −1 �
r ≡ r−1 ◦ r = 1U.
The next natural step will be to define the notion of a smooth map between two
surface rather then from a surface to an Euclidean space. The idea is the following. Let
S 1, S 2 be two surfaces In R 3 . Then any map F : S 1 → S 2 can be regarded as a map
F : S 1 → R 3 . More specifically, one can associate to F the map i ◦ F : S → R 3 , where
i : S 2 ↩→ R 3 is the inclusion.
Definition 4.7.2. Let S 1, S 2 ⊂ R 3 be two surfaces. A map F : S 1 → S 2 is called smooth
if the map F1 = i ◦ F : S 1 → R 3 is smooth.
Remarks. 1. It easy to see that any smooth between surfaces is continuous.
2. Let S 1 ⊂ R 3 a surface and G : R 3 → R 3 a diffeomorphism. Then S 2 = G(S 1) is,
also, a surface, while the map G|S 1 : S 1 → S 2 is smooth.
3. Let S 1, S 2 ⊂ R 3 two surfaces and F : S 1 → S 2 a map. Then F is smooth iff
for any a ∈ S 1, any local parameterization (U1, r1) of S 1 around a and any local
parameterization (U2, r2) of S 2 around F(a), the map
Fr1,r2
≡ r−1
2 ◦ F ◦ r1 : U1 → U2