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# Chapter 2 Gauss mappings of surfaces

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(xyXx x Xy, i.e. on the parabolic set where the Gaussian curvature K(xy) = 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(xy) = (xyf(xy)), so the Gauss map is given by N(xy) = (-fx, -fy, 1)/[(1 + (fx2 + (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.

### Example 1. The shoe surface:

X(x,y) = (xy, 1/3 x3 - 1/2 y2)

The modified Gauss mapping is Ñ(xy) = (-x2y), 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.

#### Figure 2.1

The shoe surface and its spherical image.

### Example 2. Menn's surface:

X(xy) = (xy x4 + x2y - y2)

The modified Gauss mapping is then Ñ(xy) = (-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) = ty(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(xy) =  x4 + x2y - y2 has an absolute maximum at the origin. If  > -1/4, f(xy) 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(xy) = (xy, 1/4 x4 - 1/2 y2)

The modified Gauss mapping is

Ñ(x,y) = (-x3y)

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(xy) = (xya x4 + b x2y + c y2)

also considered by Bleeker and Wilson [BlW, p. 286].

#### Figure 2.2

Menn's surface and its spherical image ( = -1/2).

#### Figure 2.3

Menn's surface and its spherical image ( = 1/4).

#### Figure 2.4

Menn's surface and its spherical image ( = -1/4).

#### Figure 2.5

The surface z = 1/4 x4 - 1/2 y2, and its spherical image.

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