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

4.9. The spherical map and the shape operator of an oriented surface 137
Theorem 4.9.1. The spherical map Γ : S → S 2 of an oriented surface S into the unit
sphere S 2 is a smooth map between surfaces.
Proof. Let a ∈ S . We choose a local parameterization (U, r) of the surface S around a,
compatible with the orientation. Clearly, since S is orientable, such a parameterization
always exists. Indeed, if we choose a parameterization (U1, r1) which is not compatible
with the orientation, i.e. we have
r ′ 1u × r′ 1v
�r ′ 1u × r′ = − n(u, v),
� 1v
then we replace the domain U1 by U− 1 , the symmetric of U1 with respect to the Ov-axis,
and the map r1(u, v) by r− 1 (u, v) = r1(−u, v). It is easy to see that the pair (U− 1 , r− 1 ) is a
parameterization of the surface, compatible with the orientation.
Now we have
(Γ ◦ r)(u, v) = Γ(r(u, v)) = Γr(u, v) = n(u, v) = r′ u × r ′
v
�r ′
u × r ′
v� .
Thus, the local representation of Γ is smooth, therefore Γ itself is smooth. �
Examples. (i) For a plane Π the spherical map is constant.
(ii) For the sphere S 2 R the map Γ : S 2 R
with x2 + y2 + z2 = R2 .
1
→ S has the expression Γ(x, y, z) = R (x, y, z),
As we saw, the tangent space to the sphere, TΓ(a)S 2 , is orthogonal to the radius
vector n(a) of the point Γ(a). On the other hand, n(a) is orthogonal to TaS . Thus, if we
identify R 3 a and R 3 Γ(a) to R3 , then the subspaces, TaS and TΓ(a)S 2 coincide. Therefore,
we may think of the differential daΓ : TaS → TΓ(a)S 2 as being, in fact, a linear operator
TaS → TaS .
Definition 4.9.1. The linear operator daΓ : TaS → TaS is called the shape operator of
the oriented surface S at the point a and it is denoted by A or Aa.
Remark. There is no general agreement regarding the definition, nor the name of the
shape operator. In some books, the shape operator carries an extra minus sign. Sometimes
it is called the Weingarten mapping, or, also the principal, or fundamental operator.
Historically, it is true, indeed, that Weingarten was the first to write down the
formulae for the differentiation of the spherical map (in other words, he was the one to
find the partial derivatives of the unit normal vectors in terms of the derivatives of the