The Binormal Surface

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|p

where 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.

Binormal vectors play a heavy role in the local geometry of surfaces immersed in R4. For instance, the classifications of hyperbolic, parabolic, or elliptic points ([MocFR]) correspond to whether a point has 2, 1, or 0 binormal vectors (actually 4, 2, or 0, since if n is a binormal vector, then so is -n).

More importantly, if we consider the Gauss map as a map from the unit normal bundle UNM to the 3-sphere S3, then this map has singularities exactly at the binormal vectors ([R], [S]). So we can consider the set of all binormals as the singularity set of the Gauss map. (Binormal vectors of space curves are also the singularities of the space curve's Gauss map, which is why we give these vectors the same name.)

The set of binormal vectors form a differentiable surface in S3, which we will call the binormal surface of M. This surface is not an immersion, and the singularities of the binormal surface can be associated to the geometry of M ([D]).

Since the binormal surface lies in S3, we can view it by stereographic projection. The binormal surface at the top of the page is from a perturbation of the complex cubic surface (z, z3), namely:

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.

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Last updated 7/19/99