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

130 Chapter 4. General theory of surfaces
that this parameterization is r1. We may assume that the support of r1 is included into
the support of ρ (otherwise, if necessary, we may shrink the domain of r1). It follows
then that the Jacobi determinant of the map ρ −1 ◦ r1 should be either always positive or
always negative on the domain of r1. On the other hand, we have, obviously, from the
chain rule, that
whence
J(ρ −1 ◦ r) = J(ρ −1 ◦ r1)J(r −1
1
◦ r),
det J(ρ −1 ◦ r) = det J(ρ −1 ◦ r1) det J(r −1
1
Now, in the right hand side, the last determinant is always positive, since the two parameterizations
are assumed to be positively equivalent. The first determinant, from the
hypothesis, is always positive or always negative. Thus, the right hand side has constant
sign. On the other hand, as we saw previously, the left hand side has opposed signs
on the two sides of the segment t = 0, whence the contradiction which shows that the
Möbius’s band is not orientable.
In the figure 4.6 we indicate how one can construct a Möbius band, from a strip of
paper. Another example of a non-orientable surface is the so-called Klein’s bottle (see
figure ??)
Definition. Let S be an oriented surface with the orientation n(a). A local parameterization
(U, r) of S is said to be compatible with the orientation n(a) if for any point
a = r(u, v) we have
n(a) = r′ u × r ′
v
�r ′
u × r ′
v�
or, which is the same, if the frame {r ′
u, r ′
v, n(a)} is right-handed.
4.7 Differentiable maps on a surface
Definition 4.7.1. Let S be a surface in R 3 . A map f : S → R k is called differentiable or
smooth if for any parameterization (U, r) of S the map f ◦ r : U → R k is smooth. The
map fr ≡ f ◦ r is called the expression of f în in the curvilinear coordinates (u, v) or the
local representation of f with respect to the parameterization (U, r).
Remarks. 1. On can define similarly the differentiability of maps defined on any
open subset of a surface S .
◦ r).
2. Any differentiable map f : S → R k is continuous, since, locally, it can be written
as a composition of continuous maps: f = f ◦ (r ◦ r −1 ) = ( f ◦ r) ◦ r −1 = fr ◦ r −1 .