Let M be a surface immersed in R4, let p be a point on M, and let n be a unit normal vector of M at p. Associated to n is a second fundamental form IIn, defined by
IIn = n . d2s|pwhere s is a local patch of M about p. As a quadratic form defined on the tangent space, the second fundamental form is independent of the parametrization. A vector n is called a binormal vector if the second fundamental form IIn is parabolic.
f(x, y) = (x, y, 3(x2 + y2) + x3 - 3xy2, 3x2y - y3)This surface has hyperbolc points when 0 < x2 + y2 < 1 and parabolic points when x2 + y2 = 1. Over each hyperbolic point, there are four sheets to the binormal surface (two on the "outside part", two on the "inside part"), and the sheets come together when we approach the parabolic curve. The origin is a special point (an imaginary inflection), and at such points two of the sheets will touch each other in a singularity (much like the evolutes of a surface in R3, see [P]). Note that there are three cuspidal edges on each sheet. These correspond to three roots of a particular cubic equation ([D]).
Projections of the perturbed z3. The arrow gives an mpeg of this surface rotated in 4-space.
NEXT: See the binormal surface colored by the same coloring scheme as the surface above.