We present the example of a polyhedral tight immersion of the projective plane with two handles in Euclidean three-space that was introduced by Kühnel and Pinkall in [3].

In the early 1960's Nicolaas Kuiper defined
and studied tight immersions of surfaces in
Euclidean three-space E^{3} in [4] and
[5].

Let M be a smooth surface with Euler characteristic chi immersed in Euclidean three-space. By definition, the total absolute curvature of M is

tau (M)=int_{M}|K|/2 do,

where K denotes the Gaussian curvature and do is the absolute value of the exterior 2-form which represents the volume. With Gauss-Bonnet's theorem and the knowledge of the following inequality for the integral over all points of positive Gaussian curvature K

int_{K>0}K/2\pi do ≥ 2,

it follows that the following inequality holds for immersions of closed surfaces in Euclidean three-space

tau (M)=int_{M}|K|/2 pi do ≥ 4-chi(M).

If equality occurs here, the immersion f: M to
E^{3} is called tight, which means
that the surface has minimal total absolute
curvature. Tightness is a generalization of
the notion of convexity. In some sense, it
means that a surface is embedded or immersed
'as convexly as possible' with respect to its
topological properties. With the introduction
of the two-piece-property (TPP), which is
equivalent to equality in the above
inequality, for surfaces in E^{3} it
was possible to examine tightness not only for
smooth surfaces but also for polyhedral ones
as it was done by Banchoff in [1]. For all
surfaces --- smooth or polyhedral --- it is
known whether they can be immersed or embedded
tightly in three-space or not. A good
collection of the known results and a huge
reference list on tightness can be found in
[2].

It was Kuiper in [5] who stated that there exists a (smooth) tight immersion of the tight projective plane with two handles in Euclidean three-space. There can also be found a very complicated construction of this immersion. It seems to be very difficult to prove rigorously that the described surface is tight although we have no doubt that the idea behind this construction is essentially correct. In this sense the example introduced in [3] and presented here, a polyhedral tight immersion of the projective plane with two handles in Euclidean three-space, was a great improvement. It was explicitely given and the tightness can be seen very easily (all the strict local supporting planes of the presented polyhedral Boy surface with two handles concentrate at the vertices of the tetrahedron which implies the tightness of the surface). Moreover the given polyhedral immersion is tightly smoothable with the algorithm given and proved in the cited paper [3] (all of the vertices are of valence three therefore the algorithm is applicable) which means that the surface can be smoothed while preserving tightness and the topological properties which proves the mentioned theorem by Kuiper more easily. Additionally the presented surface has a three-fold symmetry.

Every vertex lies in the cubic lattice
Z^{3} < E^{3} where the origin
is the triple point of the Boy surface which
is not a point of the surface. The vertices
are given as follows where the three-fold
symmetry appears as a cyclic shift of the
three coordinates (x to y to z to x)
corresponding to the cyclic shift of the
indices (1 to 2 to 3 to 1) of the presented
vertices:

A _{1}=(-1,1,0)A _{2}=(0,-1,1)A _{3}=(1,0,-1)B _{1}=(-1,1,-1)B _{2}=(-1,-1,1)B _{3}=(1,-1,-1)C _{1}=(-1,0,-1)C _{2}=(-1,-1,0)C _{3}=(0,-1,-1)D _{1}=(0,2,1)D _{2}=(1,0,2)D _{3}=(2,1,0)E _{1}=(-1,2,1)E _{2}=(1,-1,2)E _{3}=(2,1,-1)F _{1}=(-1,0,-2)F _{2}=(-2,-1,0)F _{3}=(0,-2,-1)G _{1}=(-4,-1,2)G _{2}=(2,-4,-1)G _{3}=(-1,2,-4)H _{1}=(-5,0,2)H _{2}=(2,-5,0)H _{3}=(0,2,-5)J _{1}=(2,2,-8)J _{2}=(-8,2,2)J _{3}=(2,-8,2)K=(2,2,2)

The vertices J_{1}, J_{2},
J_{3} and K span the tetrahedron which
contains a polyhedral Boy surface spanned by
the 24 vertices A_{i}, B_{i},
..., H_{i}, i=1,2,3. The 13 faces of
this Boy surface split into 5 different types
under the three-fold symmetry:

type 1: (E_{1}, G_{3}, F_{1}, C_{1}, B_{1}, A_{1}, C_{2}, B_{2}),

type 2: (A_{2}, C_{3}, F_{3}, H_{3}, D_{1}),

type 3: (B_{2}, A_{2}, D_{1}, E_{1}),

type 4: (E_{1}, D_{1}, H_{3}, G_{3}),

type 5: (H_{2}, F_{2}, G_{1}, H_{1}, F_{1}, G_{3}, H_{3}, F_{3}, G_{2}).

These faces lie in different planes, namely x=-1, x=0, z=1, y=2 and x+y+z=3, and intersect in a curve which is a closed space polygon and given by the following points:

(O L_{1}N_{2}M_{2}O L_{2}N_{3}M_{3}O L_{3}N_{1}M_{1}O),

which are defined as:

L _{1}=(1,0,0)L _{2}=(0,1,0)L _{3}=(0,0,1)M _{1}=(-1,0,0)M _{2}=(0,-1,0)M _{3}=(0,0,-1)N _{1}=(-1,0,1)N _{2}=(1,-1,0)N _{3}=(0,1,-1)O=(0,0,0).

The presented geometry is built up of the faces given above with the given coordinates.

The Euler-characteristic chi of the Boy surface is easily computed to be chi=24-36+13=1. After gluing it together with the tetrahedron in the way as it can be seen in the model (the faces of type 4 indicate the way, the tetrahedron and the Boy surface are glued together) we get chi=28-42+11=-3. Note that the faces of the tetrahedron are not contractible, with the result that they do not make a contribution to the Euler-characteristic. Therefore the shown surface is a polyhedral tight immersion of the projective plane with two handles as stated.

Model produced with: JavaView v.2.21

Keywords | Tight surface; Total absolute curvature; Projective plane; Euclidean 3-space | |

MSC-2000 Classification | 53C42 | |

Zentralblatt No. | 05264894 |

- T. Banchoff:
*Tightly embedded 2-dimensional polyhedral manifolds*, Amer. J. Math.**87**(1965), 462--472, . - T. Banchoff and W. Kühnel:
*Tight submanifolds, smooth and polyhedral*, in T. Cecil and S-s. Chern (Eds.): Tight and taut submanifolds, Cambridge University Press (1984), 51--118, . - W. Kühnel and U. Pinkall:
*Tight smoothing of some polyhedral surfaces*, in A. Dold and B. Eckmann (Eds.): Global Differential Geometry and Global Analysis, Springer Verlag (1984), 227--239, . - N. Kuiper:
*On surfaces in euclidean three-space*, Bull. Soc. Math. Belg.**12**(1960), 5--22, . - N. Kuiper:
*Convex immersions of closed surfaces in E^3*, Comment. Math. Helv.**35**(1961), 85-92, .

- Master File: projective_plane_with_2_handles.jvx
- Applet File: projective_plane_with_2_handles.jvx
- Preview: projective_plane_with_2_handles.jpg
- Print File: projective_plane_with_2_handles.ps
- Other: projective_plane_with_2_handles.pdf

Submitted: Wed May 14 14:50:05 GMT 2003.

Revised: Fri Aug 20 10:10:50 CEST 2004.

Accepted: Sat Sep 4 11:13:43 GMT 2004.

University StuttgartMarc-Oliver Otto

Department of mathematics

Institute of geometry and topology

Pfaffenwaldring 57

70550 Stuttgart

Germany

Bulach@mathematik.uni-stuttgart.de

http://www.mathematik.uni-stuttgart.de/igt/LstGeo/Bulach/

University Stuttgart

Department of mathematics

Institute of geometry and topology

Pfaffenwaldring 57

70550 Stuttgart

Germany

Otto@mathematik.uni-stuttgart.de

http://www.mathematik.uni-stuttgart.de/igt/LstDiffgeo/Otto/