For an embedding X : U -> R3 of a parameter domain U in R2 into Euclidean 3-space, the Gauss mapping N : U-> S2 sends each point (x, y) of U to the unit normal N = (Xx x Xy) / |Xx x Xy|. The Gauss mapping is singular precisely when 0 = Nx x Ny = K(x, y) Xx x Xy, i.e. on the parabolic set where the Gaussian curvature K(x, y) = 0.
In the terminology of Whitney [Wh] , the Gauss mapping N is good if the gradient of K is never zero on the parabolic set. If N is good, then the parabolic set is a smooth curve (x(t), y(t)). The image N(t) of this curve under the Gauss map is singular precisely when N'(t) = 0. If N is good, then N is excellent if N'(t) = 0 implies N"(t) 0. This ensures that the singularities of the curve N(t) are cusps. Finally, if N is excellent, then N is in general position if the image of N(t) has no triple points or self-tangencies, and no cusp point of N(t) coincides with another image point of N(t).
Whitney proved that a map of surfaces is excellent if and only if its singularities are all equivalent (by smooth changes of coordinates) to folds or cusps. Furthermore, a map of surfaces is stable if and only if it is excellent and in general position. (For a precise definition of stability and a discussion of Whitney's theorem, see [A1] and [GolG].)
We begin our investigation of the singularities of the Gauss mapping with a collection of key examples which exhibit all of the geometric phenomena which we shall associate with these singularities.
Our first three examples are function graphs of the form X(x, y) = (x, y, f(x, y)), so the Gauss map is given by N(x, y) = (-fx, -fy, 1)/[(1 + (fx) 2 + (fy)2]1/2
We can study the singularities of the Gauss mapping more easily in this case by projecting centrally from the origin to the plane z = 1 to get (-fx, -fy, 1). We then project to the xy-plane to get the composed mapping
Ñ(x,y) = (-fx, -fy)
Since the image of N is contained in the upper hemisphere, and central projection is a diffeomorphism from the upper hemisphere to the plane z = 1 , the modified Gauss mapping Ñ will have the same singularities as N. In particular N is singular precisely when the Jacobian matrix
has rank less than two, i.e. when the discriminant = (fxy)2 - fxx fyy is zero.
X(x,y) = (x, y, 1/3 x3 - 1/2 y2)
The modified Gauss mapping is Ñ(x, y) = (-x2, y), and the parabolic curve is obtained by solving 0 = = (fxy)2 - fxx fyy = 2 x. Since grad = (2, 0) 0, the mapping Ñ is good. The parabolic curve can be parametrized by x(t) = 0, y(t) = t. The modified Gauss mapping restricted to the parabolic curve is Ñ(t) = (0, t), with Ñ'(t) = (0, 1) 0, so Ñ is excellent. Thus the Gauss map is stable, with a simple fold along the parabolic curve.
The shoe surface and its spherical image.
X(x, y) = (x, y, x4 + x2y - y2)The modified Gauss mapping is then Ñ(x, y) = (-4 x3 - 2 xy, -x2 + 2y) and the parabolic curve is obtained by solving 0 = = (24 + 4) x2 + 4 y. Since grad = ((48 + 8) x, 4) 0, the mapping Ñ is good for all . The parabolic curve can be parametrized by x(t) = t, y(t) = -(6 + 1) t2, so Ñ restricted to the parabolic curve is
Ñ(t) = (2(4 + 1) t3, -3 (4 + 1) t2)
and Ñ'(t) = 0 implies Ñ"(t) 0 , if -1/4. Therefore the Gauss map is stable if -1/4, with a cusp at the origin. For = -1/4, the entire parabolic curve is sent to a single point by the Gauss map. (This is similar to the situation which occurs at the top rim of a torus of revolution -cf. examples 4 and 5 below.) If < -1/4, f(x, y) = x4 + x2y - y2 has an absolute maximum at the origin. If > -1/4, f(x, y) has a topological saddle point at the origin. The case = 0 was first studied by Michael Menn.
A related example of a surface with an unstable Gauss map is
X(x, y) = (x, y, 1/4 x4 - 1/2 y2)The modified Gauss mapping is
Ñ(x,y) = (-x3, y)
and = 3 x2, so the parabolic curve is the line x = 0. Since grad = 6x is zero on the parabolic curve, the Gauss map is not good. Both this surface and Menn's surface occur in the 3-parameter family
X(x, y) = (x, y, a x4 + b x2y + c y2)also considered by Bleeker and Wilson [BlW, p. 286].