Clifford parallel

A straight line in an elliptic space that stays at a constant distance from a given (base) straight line. Through each point lying outside a given line and outside its polar line there pass two Clifford parallels to the given line. The surface formed by rotating a Clifford parallel about its base line is called a Clifford surface. A Clifford surface has constant zero Gaussian curvature.

W. Clifford (1873) was the first to show the existence of Clifford surfaces.

References

[1]	S.A. Bogomolov, "An introduction to Riemann's non-Euclidean geometry" , Leningrad-Moscow (1934) (In Russian)
[2]	B.A. Rozenfel'd, "Non-Euclidean spaces" , Moscow (1969) (In Russian)

Comments

Let be -dimensional Euclidean space, and its associated projective space of all straight lines through the origin. For let be the angle between the lines and in . Then with this metric is called the elliptic space associated with . The topology induced by this metric is the usual one, i.e. the quotient topology of . The article above deals with the case .

The (absolute) polar line to the line through two points and of is the line of all points such that , where denotes the usual inner product.

The notion of Clifford parallelism is also considered on the -fold covering of , [a2].

References

[a1]	F. Klein, "Vorlesungen über nichteuklidische Geometrie" , Springer (1928)
[a2]	M. Berger, "Geometry" , II , Springer (1987) pp. 84