A sphere immersed into four-dimensional space which intersects itself at exactly one point.

We immerse the 2-sphere into **R**^{4} by the equation

(*x*, *y*, *x*^{2} + *xz*, *yz*)
where
*x*^{2} + *y*^{2} + *z*^{2} = 1.

The main importance of this surface is that it is a simple example
of a manifold that has a normal bundle with nontrivial characteristic
classes, and it is good example of the connection between the geometry
of a surface and its normal Euler number. Specifically, it is a
closed surface in **R**^{4} that intersects itself at
exactly one point (namely the north and south poles of the sphere are
mapped to the origin in **R**^{4}). In turn, this gives a
simple example of a oriented surface with a nonzero normal Euler
number (the integral of the normal curvature of the surface, also the
obstruction to a global nonzero normal vector field). Specifically,
the self-intersecting sphere has normal Euler number equal to 2.

The normal Euler number appears in various counting formulae for
surfaces: the algebraic sum of zeros of a normal vector field (one
definition of the normal Euler number), the algebraic number of pinch
points of a projection of the surface into **R**^{3}, the
algebraic number of complex points [1, 2], and the author's bitangency
formula [3]. This sphere is an important example for all these
formulae, having not only a nonzero normal Euler number but also
because its equation is simple enough to allow computation of other
terms in the formulae.

The surface presented is a modification of a a more symmetric
surface (*x*, *y*, *xz*, *yz*) found in [1], which
in turn is a generalization of the figure eight curve (*x*,
*xy*), *x*^{2}+*y*^{2}=1.
Unfortunately, this surface has too much symmetry (all circles of
latitude on the sphere map to circles on the surface), and so some
generic conditions are lost. However, a modification in the quadratic
terms will preserve the important geometric features of the surface
while at the same time breaking up the symmetry, as well as allowing
one to arbitrarily set the second-order local geometry at certain
points on the surface.

The bitangency formula (a higher dimensional generalization of a
formula for plane curves [4]) counts the algebraic number of
bitangencies (pairs of points which share a common tangent line) and
relates this number to the number of self intersections, the normal
Euler number, and a "diagonal contribution" which comes from the local
geometry of the surface [3]. When one projects the surface into
**R**^{3} down the direction of a bitangency, the pair of
points are mapped to a pair of overlapping pinch points. In our
model, this can be seen in two places, marking two bitangencies on the
surface. There are two other bitangencies on this surface, marked by
the two lines on the picture above.

Model produced with: Geomview 1.6.1 with the Centerstage module

Keywords | normal Euler number; complex point; bitangency | |

MSC-2000 Classification | 53A07 (57R20) | |

Zentralblatt No. | 05264882 |

- T. Banchoff and F. Farris:
*Tangential and normal Euler numbers, complex points, and singularities of projections for oriented surfaces in four-space*, Pac. J. Math**161**(1993), 1-24. - S.-S. Chern and E. H. Spanier:
*A theorem on orientable surfaces in four-dimensional spaces*, Comm. Hath. Helv.**25**(1951), 205-209. - D. Dreibelbis:
*A bitangency theorem for surfaces in four dimensions*, Quart. J. Math (To Appear). - F. Fabricius-Bjerre:
*On the double tangents of plane curves*, Math. Scand.**36**(1962), 83-96.

- Master File: Sphere_Master.jvx
- Applet File: Sphere_Applet.obj
- Preview: Sphere_Preview.jpg
- Other: Sphere_Cut.jvx

Sphere-Cut.jvx contains the surface with alternate bands removed, useful for studying the cross sections.

Submitted: Mon Jun 18 19:06:54 MDT 2001.

Revised: Mon May 6 20:56:11 GMT 2002.

Accepted: Mon Sep 16 12:57:20 CEST 2002.

University of North Florida

Department of Mathematics and Statistics

4567 St. John's Bluff Rd.

Jacksonville, FL 32224

ddreibel@unf.edu

http://www.unf.edu/~ddreibel