The following three theorems will be illustrated in this section by the five examples of section two.
Theorem 3.1 Let U be an open set in the plane and let X : U-> R^{3} be a smooth immersion. If P in U is a cusp of the Gauss map of X, then the following statements (a)-(h) are true. Conversely, if the Gauss map of X is stable, and any of the following conditions holds, then P is a cusp of the Gauss mapping.
Our final two characterizations of Gaussian cusps involve notions which aren't defined for an arbitrary immersion.
Theorem 3.2 Let X : U->R^{3} be a smooth immersion of the planar open set U , and let P be a parabolic point of X . If P is a cusp of the Gauss map, and the image of the parabolic curve under X has nonzero curvature at P , then
Theorem 3.2, except for the last statement, is due to Bleeker and Wilson [BlW].
Theorem 3.3 Let X : U->R^{3} be a smooth immersion of the planar open set U , and let P be a parabolic point of X . If P is a cusp of the Gauss map, and the asymptotic direction map along the parabolic curve is regular at P , then
Conversely, if the Gauss map is stable, and the asymptotic direction map is regular at P , and (j) holds, then P is a cusp of the Gauss map. Furthermore, the set of immersions such that the Gauss map is stable and the asymptotic direction map is regular at each Gaussian cusp is open and dense in the space of all immersions of U in R^{3}.
The topology on the space of smooth immersions of U in R^{3} is the Whitney C-infinity topology [GolG, p. 42].
Conditions (b), (c), and (i) are related to the contact of the immersion X with planes in R^{3} , or to singularities of projections to lines in R^{3} . Conditions (d), (e), and (f) reflect the contact of X with spheres in R^{3} , or singularities of the distance-squared function from points of R^{3} . Conditions (g), (h), and (j) are related to the contact of X with lines in R^{3} , or to singularities of projections of X to planes in R^{3}.
The proofs of these theorems, in chapters 5, 6, and 7, will actually establish global versions, for immersions in R^{3} of any smooth surface.
We now discuss the ten properties (a)-(j), using the five examples of chapter two.
If P is a cusp of the Gauss map N of X then there are coordinate charts about P in U and about N(P) in S^{2} such that in these coordinates N takes (x, y) to (x, y^{3} - xy). The curve x = 3y^{2} of singular points of N is the parabolic curve of X near P = (0, 0). Its image by N is the cuspidal cubic 4u^{3} = 27v^{2}. If 4u^{3} > 27v^{2} there are exactly three points Q_{1}, Q_{2}, Q_{3} such that N(Q_{i}) = (u, v), and exactly two of these points Q_{i} = (x_{i}, y_{i}) satisfy x_{i} < 3y_{i}^{2}. If 4u^{3} < 27v^{2} there is exactly one point Q = (x, y) such that N(Q) = (u, v); this point has x < 3y^{2}. So there are two distinct types of Gaussian cusps:
If P is a cusp of the Gauss map N of X , then P is elliptic [hyperbolic] if and only if there exist distinct elliptic [hyperbolic] points Q_{1} and Q_{2} arbitrarily close to P such that N(Q_{1}) = N(Q_{2}).
Menn's surface (example 2) has an elliptic Gaussian cusp if < -1/4 , and a hyperbolic Gaussian cusp if > -1/4.
Property a) If the Gauss mapping is good then its Jacobian has rank one at each parabolic point P . The kernel of the Jacobian map is the zero principal curvature direction at P. Consider a curve (x(t), y(t)) in the parameter domain U with Gaussian image N(t). Let P = (x(0), y(0)) and V = (x'(0), y'(0)). The curvature of X in the direction V at P is zero if and only if N'(0) = 0. If (x(t), y(t)) is the parabolic curve of X, and N is excellent, then N'(0) = 0 if and only if P is a cusp of N. This implies theorem 3.1 (a).
The modified Gauss mapping of the shoe surface (example 1) has Jacobian matrix
so the zero principal curvature direction along the parabolic curve (0, t) is spanned by the constant vector (1, 0), and (a) holds for no parabolic point.
The modified Jacobian matrix of Menn's surface is
so the zero principal curvature direction along the parabolic curve (t, -(1 + 6 )t^{2}) is spanned by the vector (1, t). If -1/4, this vector is tangent to the parabolic curve if and only if t = 0, where the Gauss map has a cusp. If = -1/4, the vector (1, t) is tangent along the entire parabolic curve, and the Gauss map is not excellent.
The two types of Gaussian cusps can be distinguished from the viewpoint of (a) as follows. If P is a cusp of the Gauss map of X, let l be the line of curvature of X which is tangent to the parabolic curve at P . If P is elliptic [hyperbolic], then l - {P} is contained in the elliptic [hyperbolic] region of U (cf. [BlW, p. 287]).
Property b) A plane M is tangent to the surface X : U->R^{3} at Q in U if and only if Q is a critical point of the composition of X with orthogonal projection to the line l through the origin normal to M. Then the line l contains the Gaussian spherical image N(Q).
For a unit vector V in R^{3} , let _{V} : U->R be the composition of X with the orthogonal projection to the line spanned by V:
_{V}(P) = X(P) ^{.} V
A restatement of (b) is that the height function _{N(P)} has a degenerate critical point at P, which splits up into three distinct critical points Q_{1}, Q_{2}, Q_{3} near P for certain height functions _{V} with V arbitrarily near N(P) . We will show in chapter 5 that these three critical points are nondegenerate if P is a cusp of the Gauss map N. So (b) is equivalent to the statement that the critical point P of the height function _{N(P)} has Milnor number three (cf. [C2]).
The two types of Gaussian cusps are distinguished from this viewpoint by their Morse indices. P is an elliptic [hyperbolic] cusp of the Gauss map N if and only if P is an extremum [saddle] of _{N(P)}. P splits up into three nondegenerate critical points: two maxima (or two minima) and a saddle [two saddles and a maximum (or minimum)]. Note that _{N(Q)} has a nondegenerate critical point at Q which is an extremum [saddle] if and only if the Gaussian curvature of X at Q is positive [negative].
Now consider a function graph X(x, y) = (x, y, f(x, y)), with P = (0, 0) and N(P) = (0, 0, 1). Then _{N(P)}(x, y) = f(x, y). The family of height functions _{V} : U->R, _{V}(x, y) = X(x, y) ^{.} V, with V = (a, b, c), a^{2} + b^{2} + c^{2} = 1, c > 0, is equivalent to the family _{V} with a and b arbitrary and c = 1. Let _{(a, b)} = _{(a, b, 1)}. Then
_{V}(x, y) = f(x, y) + ax + by,
a linear perturbation of f(x, y). And Q is a critical point of _{(a, b)} if and only if (a, b) = Ñ(Q), where Ñ is the modified Gauss map of X (defined in chapter 2). The Hessian matrix of second partials of _{V} equals the Hessian matrix of f(x, y), which is the negative of the Jacobian matrix of Ñ. Therefore Q is a degenerate critical point of _{(a, b)} if and only if Q is a parabolic point of X.For the shoe surface we have the perturbation
with gradient (x^{2} + a, -y + b). If a > 0, _{(a, b)} has no critical points. If a < 0, _{(a, b)} has critical points ((-a)^{1/2}, b) and (-(-a)^{1/2}, b) with Hessian matrices
respectively, so ((-a)^{1/2}, b) is a saddle and (-(-a)^{1/2}, b) is a maximum. (If a = 0, _{(a, b)} has a degenerate critical point at (0, b).) Therefore f(x, y) has Milnor number 2 at P = (0, 0). For Menn's surface we have
with gradient (4x^{3} + 2 xy + a, x^{2} - 2 y + b). So (x, y) is a critical point of _{(a, b)} if and only if both
(Note that the locus of zeros of the discriminant is precisely the image of the parabolic curve of X by the modified Gauss map Ñ.) If < -1/4, then f(x, y) = 0 only if (x, y) = (0, 0), so f has an absolute maximum at (0, 0), and P = (0, 0) is an elliptic Gaussian cusp. If > -1/4, then the locus f(x, y) = 0 is two tangent parabolas, and f has a topological saddle point at (0, 0), so P is a hyperbolic Gaussian cusp. Theorem 3.1(b) will be proved in chapter 5.