Gauss mappings of surfaces

For an embedding
**X** : *U* -> **R**^{3} of a
parameter domain *U* in **R**^{2} into Euclidean
3-space, the Gauss mapping
**N** : *U*-> S^{2} sends each point
(*x*, *y*) of *U* to the unit normal
**N** = (**X**_{x} x **X**_{y}) / |**X**_{x} x **X**_{y}|.
The Gauss mapping is singular precisely when
0 = **N**_{x} x **N**_{y} = *K*(*x*, *y*) **X**_{x} x **X**_{y},
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*) = (-*f*_{x}, -*f*_{y}, 1)/[(1 + (*f*_{x}) ^{2} + (*f*_{y})^{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
(-*f*_{x}, -*f*_{y}, 1).
We then project to the *xy*-plane to get the composed mapping

**Ñ**(*x*,*y*) = (-*f*_{x}, -*f*_{y})

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
= (*f*_{xy})^{2} - *f*_{xx} f_{yy}
is zero.

**X**(*x*,*y*) = (*x*, *y*, 1/3 *x*^{3} - 1/2 *y*^{2})

The modified Gauss mapping is
**Ñ**(*x*, *y*) = (-*x*^{2}, *y*),
and the parabolic curve is obtained by solving
0 = = (*f*_{xy})^{2} - *f*_{xx} f_{yy} = 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*, *x*^{4} + *x*^{2}*y* - *y*^{2})

**Ñ**(*t*) = (2(4 + 1) *t*^{3}, -3 (4 + 1) *t*^{2})

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*) = *x*^{4} + *x*^{2}*y* - *y*^{2}
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 *x*^{4} - 1/2 *y*^{2})

**Ñ**(*x*,y) = (-*x*^{3}, *y*)

and = 3 *x*^{2}, so the
parabolic curve is the line *x* = 0. Since
grad = 6*x* 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* *x*^{4} + *b* *x*^{2}*y* + *c y*^{2})