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

4.7. Differentiable maps on a surface 131
Figure 4.3: Construct your own Möbius band!
Examples. a) Any constant map f : S → R k : a → A0, A0 ∈ R k is smooth, because
its local representation with respect to any local parameterization of S is, equally, a
constant map, hence it is differentiable.
b) If F : R 3 → R k is a smooth map, then, for any surface S the map f = F|S : S →
R k is a smooth map. Indeed, for any local parameterization (U, r) of S the local
representation of f is fr = F ◦ r, where F and r are smooth maps, in the usual sense.
In particular, the orthogonal projections of a surface S on the coordinate axes and
planes are, all of them, smooth maps.
c) The inclusion i : S → R 3 : a → a is smooth, since for any local parameterization
(U, r) of S , the local representation of i is ir = i◦r = r. (In fact, i is just the restriction
of the identity map, 1 R 3 and, therefore, we can also apply the previous example).
Apparently, it is quite difficult to check whether a map defined on a surface is smooth,