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