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 |
Clifford parallel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Clifford_parallel&oldid=39863