Chapter 3

Characterizations of Gaussian cusps

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-> R³ 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.

a)

A zero principal curvature direction of X is tangent to the parabolic curve of X at P.

b)

For each  > 0 there exist three distinct points Q₁, Q₂, Q₃ in U such that |P - Q_i| <  for i = 1, 2, 3, and the tangent planes to X at Q₁, Q₂, Q₃ are parallel.

c)

For each  > 0 there exist two distinct points Q₁, Q₂ in U such that |P -Q_i| <  for i = 1, 2, and the tangent planes to X at Q₁ and Q₂ are equal.

d)

P is a ridge point of X as well as a parabolic point of X , and the principal curvature associated with the ridge is zero at P.

e)

Given  > 0 and d > 0, there exists a point Q in U and a D > d such that |P - Q| <  and Q is a swallowtail point of the parallel surface to X at distance D .

f)

For any point A in R³ which is not on the tangent plane to X at P, the point P is a swallowtail point of the pedal surface of X from A.

g)

P is a parabolic point of X , and a line in R³ has order of contact greater than two with X at P.

h)

P is a parabolic point of X , and given  > 0 there exists Q in U such that |P - Q| <  and Q is an inflection point of an asymptotic curve of X.

Our final two characterizations of Gaussian cusps involve notions which aren't defined for an arbitrary immersion.

Theorem 3.2 Let X : U->R³ 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

i)

The osculating plane at P of the parabolic image curve of X is the tangent plane of X at P.

Theorem 3.2, except for the last statement, is due to Bleeker and Wilson [BlW].

Theorem 3.3 Let X : U->R³ 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

j)

P is an inflection point of the asymptotic direction map of X.

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 R³.

The topology on the space of smooth immersions of U in R³ 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 R³ , or to singularities of projections to lines in R³ . Conditions (d), (e), and (f) reflect the contact of X with spheres in R³ , or singularities of the distance-squared function from points of R³ . Conditions (g), (h), and (j) are related to the contact of X with lines in R³ , or to singularities of projections of X to planes in R³.

The proofs of these theorems, in chapters 5, 6, and 7, will actually establish global versions, for immersions in R³ 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 S² such that in these coordinates N takes (x, y) to (x, y³ - xy). The curve x = 3y² of singular points of N is the parabolic curve of X near P = (0, 0). Its image by N is the cuspidal cubic 4u³ = 27v². If 4u³ > 27v² there are exactly three points Q₁, Q₂, Q₃ such that N(Q_i) = (u, v), and exactly two of these points Q_i = (x_i, y_i) satisfy x_i < 3y_i². If 4u³ < 27v² there is exactly one point Q = (x, y) such that N(Q) = (u, v); this point has x < 3y². So there are two distinct types of Gaussian cusps:

(i)

Elliptic cusp. If x < 3y² then (x, y) is elliptic (K(x, y) > 0), and if x > 3y² then (x, y) is hyperbolic (K(x, y) < 0).

(ii)

Hyperbolic cusp. If x < 3y² then (x, y) is hyperbolic, and if x > 3y² then (x, y) is elliptic.

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 Q₁ and Q₂ arbitrarily close to P such that N(Q₁) = N(Q₂).

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 )t²) 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->R³ 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 R³ , 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 Q₁, Q₂, Q₃ 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), a² + b² + c² = 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,

For the shoe surface we have the perturbation

(i)

y = -(4x³ + a)/2x, and

(ii)

(4 + 1) x³ + bx + a = 0,