Elliptic geometry

A geometry in a space with a Riemannian curvature that is constant and positive in any two-dimensional direction. Elliptic geometry is a higher-dimensional generalization of the Riemann geometry.

Comments

Thus, elliptic geometry is the geometry of real projective space endowed with positive sectional curvature (i.e. the geometry of the sphere in $ \mathbf R ^ {n} $ with antipodal points, or antipodes, identified). An exposition of it is given in [a1], Chapt. 19; generalizations are given in [a2]. Some details follow.

Let $ E $ be an $ ( n+ 1 ) $- dimensional Euclidean space and $ P = \mathbf P ( E) $ the associated projective space of all straight lines through the origin. For $ L , L ^ \prime \in P $ let $ d ( L , L ^ \prime ) \in [ 0 , \pi / 2 ] $ be the angle (in the Euclidean sense) between the lines $ L $ and $ L ^ \prime $ in $ E $. If $ l $ and $ l ^ \prime $ are two lines in $ P $ intersecting in $ L $, then the angle between $ l $ and $ l ^ \prime $ is the angle in $ [ 0 , \pi /2 ] $ between the corresponding planes $ l $ and $ l ^ \prime $ in $ E $( which intersect in the line $ L $). The space $ P $ with this metric (and this notion of angle) is called the elliptic space associated with $ E $. It is of course closely related to the spherical geometry of $ S ( E) = \{ {x \in E } : {\| x \| = 1 } \} $, being in fact a quotient. The topology induced by the metric is the usual one.

Consider for the moment the spherical geometry of $ S ^ {2} $, i.e. the lines are great circles. Take e.g. the equator. Then all lines in $ S ^ {2} $ perpendicular to the equator meet in the North and South poles, the polar points of the equator. Identifying antipodal points one obtains $ \mathbf P ( \mathbf R ^ {3} ) $, in which therefore for every line $ l $ there is unique point point $ A $, the (absolute) polar of $ l $ through which every line perpendicular to $ l $ passes. Conversely, to every point $ A $ of $ \mathbf P ( \mathbf R ^ {3} ) $ there corresponds an (absolute) polar line.

This generalizes. Let $ d \subset P $ be an $ r $- dimensional plane in $ P $, then the (absolute) polar of $ d $ in $ P $ is the plane $ e $ of dimension $ s = n - r - 1 $ consisting of all points $ x = ( x _ {0} : x _ {1} : \dots : x _ {n} ) $ such that for all $ y = ( y _ {0} : y _ {1} : \dots : y _ {n} ) \in d $, $ \langle x , y \rangle = \sum x _ {i} y _ {i} = 0 $. Thus, for $ \mathbf P ( \mathbf R ^ {4} ) $ the polar of a line is a line.

References

[a1]	M. Berger, "Geometry" , II , Springer (1987)
[a2]	H. Busemann, "Recent synthetic differential geometry" , Springer (1970)