The Projective Plane with Two Handles:

To generate tight immersions of surfaces with odd Euler characteristic, Kuiper began by looking for a simple immersion of the real projective plane. For a tight surface, the height function for almost all directions is polar (that is, it has one minimum, one maximum, and the smallest possible number of saddles). Since the Euler characteristic can be computed by adding the number of maxima and minima and subtracting the number of saddles, a polar function on the real projective plane (which has Euler characteristic 1) would have exactly one maximum, one minimum, and one saddle.

Kuiper described an immersion of the projective plane by giving the levels sets for different heights of a polar height function, as in the following picture:

Here, the arrows point to the three critical points. The difficult transition to visualize is the one between the third and fourth levels from the top. This is explained in more detail at the right, where the red segment is moved across the upper loop to the purple position. The triple point is formed as the red line moves through the crossing formed by the upper loop. The complete surface can be obtained by "filling in" bands of surface between the different levels.

A smooth tight surface has the property that all its positive curvature must be on its convex envelope, so everything inside the convex envelope must have negative or zero curvature [More]. Kuiper's idea was to try to build the projective plane so that, if he cut off the maximum at the top and the minimum at the bottom, the rest would be negatively curved; this would form the central core of a tight immersion of a projective plane with a handle (it would be pasted into a big sphere with two disks removed, which would form a handle from the top to the bottom, and would make the surface tight).

Kuiper tried to accomplish this by using pieces of ruled surfaces to connect the level sets that he had drawn, and since every point on a ruled surface has a direction with zero curvature, the curvature is everywhere non-positive. The only possible problem would occur at the interfaces between two pieces of ruled surface where they join along one of the level sets.

Unfortunately, Kuiper could not connect the ruled surfaces without introducing some positive curvature between the points A and B in the diagram. To overcome this problem, he introduced an additional handle (inset at right) that attached to the outer shell, thus removing the area of positive curvature from the central core.

This model serves as the basis for the tight immersions of the remaining non-orientable surfaces: handles can be added to this surface as before in order to obtain tight immersions of any surface with odd Euler characteristic less than -3.

[Right] There is no tight projective plane
[Left] Non-orientable tight surfaces
[Up] Kuiper's initial question
[Up] The level sets of the polyhedral solution

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7/20/94 -- The Geometry Center