# 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.

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.