Real Projective Plane

The closed topological Manifold, denoted $\Bbb{R}P^2$, which is obtained by projecting the points of a plane $E$ from a fixed point $P$ (not on the plane), with the addition of the Line at Infinity, is called the real projective plane. There is then a one-to-one correspondence between points in $E$ and lines through $P$. Since each line through $P$ intersects the sphere $\Bbb{S}^2$ centered at $P$ and tangent to $E$ in two Antipodal Points, $\Bbb{R}P^2$ can be described as a Quotient Space of $\Bbb{S}^2$ by identifying any two such points. The real projective plane is a Nonorientable Surface.

The Boy Surface, Cross-Cap, and Roman Surface are all homeomorphic to the real projective plane and, because $\Bbb{R}P^2$ is nonorientable, these surfaces contain self-intersections (Kuiper 1961, Pinkall 1986).

See also Boy Surface, Cross-Cap, Nonorientable Surface, Projective Plane, Roman Surface

References

Geometry Center. ``The Projective Plane.'' http://www.geom.umn.edu/zoo/toptype/pplane/.

Gray, A. ``Realizations of the Real Projective Plane.'' §12.5 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 241-245, 1993.

Klein, F. §1.2 in Vorlesungen über nicht-euklidische Geometrie. Berlin, 1928.

Kuiper, N. H. ``Convex Immersion of Closed Surfaces in $E^3$.'' Comment. Math. Helv. 35, 85-92, 1961.

Pinkall, U. Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 64-65, 1986.