Real Projective Plane

The closed topological Manifold, denoted , which is obtained by projecting the points of a plane from a fixed point (not on the plane), with the addition of the Line at Infinity,is called the real projective plane. There is then a one-to-onecorrespondence between points in and lines through . Since each line through intersects the sphere centered at and tangent to in two Antipodal Points, can be described as a Quotient Spaceof 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 is nonorientable, these surfaces contain self-intersections (Kuiper 1961, Pinkall 1986).

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 .'' 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.

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