Nonorientable Surface

A nonorientable surface, such as the Möbius strip or Klein bottle (Gray 1997, pp. 322-323), is a surface on which there exists a closed path such that the directrix is reversed when moved around this path. The real projective plane is also a nonorientable surface. Models of it include the Boy surface, Roman surface, and the capped model of the cross-cap (Pinkall 1986).

The following construction gives maps of the real projective plane into (Banchoff 1984, Pinkall 1986). Choose three homogeneous polynomials of positive even total degree and consider the map

(1)

Then restrict , , and to the sphere by writing

			(2)
			(3)
			(4)

with and . Since the total degrees are even, . Thus antipodal points on the sphere have the same image, and identifying antipodal points on the boundary of this hemisphere gives a map of the real projective plane into . The resulting map need not be an immersion.

In three dimensions, there is no unbounded nonorientable surface which does not intersect itself (Kuiper 1961, Pinkall 1986).