No Tight Projective Plane:

To show that there is no tight immersion of the real projective plane, Kuiper used the fact that a tight immersion of a surface M can be decomposed into two components, M+ and M-, with the following properties: [More]

  1. The curvature is non-negative on M+ and non-positive on M-,
  2. The image of M+ is an embedding and is equal to the convex envelope of the image of M minus a finite number of planar, convex disks,
  3. The boundary of each of these disks is the image of a curve in M that doesn't bound a region in M.

The boundary curves mentioned in (3) are called top cycles, and they play a key role in understanding tight immersions. Note that a top cycle is an embedded, planar, convex curve, and that every top cycle has an orientable neighborhood [More].

If a tight immersion of a connected surface has no top cycles, then it is necessarily a sphere, since in this case the image of the M+ region is all of the convex envelope (and there is no M- region). So a tight immersion of the projective plane must have at least one top cycle.

There is only one class of embedded curves on the projective plane that do not bound regions; but curves in this class have non-orientable neighborhoods (the neighborhood is a Möbius band, as in the diagram below), and so they can not be top cycles.

This is a contradiction, so there can be no tight immersion of the real projective plane.

[Right] There is no tight Klein bottle
[Left] Non-orientable tight surfaces
[Up] Kuiper's initial question

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7/21/94 -- The Geometry Center