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 $E$ be $(n+1)$-dimensional Euclidean space, and $P = \mathbf{P}(E)$ its associated projective space of all straight lines through the origin. For $L,L' \in P$ let $d(L,L') \in [0,\pi/2]$ be the angle between the lines $L$ and $L'$ in $E$. Then $P$ with this metric is called the elliptic space associated with $E$. The topology induced by this metric is the usual one, i.e. the quotient topology of $E \rightarrow P$. The article above deals with the case $n=3$.

The (absolute) polar line to the line $\ell$ through two points $x = (x_0:x_1:x_2:x_3)$ and $y = (y_0:y_1:y_2:y_3)$ of $\mathbf{P}(\mathbf{R}^4)$ is the line of all points $z = (z_0:z_1:z_2:z_3)$ such that $\langle x,z \rangle = \langle y,z \rangle = 0$, where $\langle {\cdot},{\cdot} \rangle$ denotes the usual inner product.

The notion of Clifford parallelism is also considered on the $2$-fold covering $S^3$ of $\mathbf{P}(\mathbf{R}^4)$, [a2].

References

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