# The Binormal Surface

Let *M* be a surface immersed in **R**^{4}, 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 II_{n}, defined by

II_{n} = **n** ^{.} *d*^{2}**s**|_{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
II_{n} is parabolic.

Binormal vectors play a heavy role in the local geometry of surfaces
immersed in **R**^{4}. 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 *S*^{3}, 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
*S*^{3}, 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 *S*^{3}, 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*, *z*^{3}), namely:
*f*(*x*, *y*) = (*x*, *y*,
3(*x*^{2} + *y*^{2}) + *x*^{3} - 3*x**y*^{2}, 3*x*^{2}*y* -
*y*^{3})

This surface has hyperbolc points when 0 < *x*^{2} +
*y*^{2} < 1 and parabolic points when
*x*^{2} + *y*^{2} = 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 **R**^{3}, 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 *z*^{3}. 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.

BACK to Dan's Research Page.

Last updated 7/19/99