- The curvature is non-negative on
*M+*and non-positive on*M-*, - 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, - 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
.

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.

* 7/21/94 dpvc@geom.umn.edu -- *

*The Geometry Center*