Editing Complex projective space (section)

===Geodesics===
Through any two points ''p'', ''q'' in complex projective space, there passes a unique ''complex'' line (a '''CP'''<sup>1</sup>).  A [[great circle]] of this complex line that contains ''p'' and ''q'' is a [[geodesic]] for the Fubini&ndash;Study metric.  In particular, all of the geodesics are closed (they are circles), and all have equal length.  (This is always true of Riemannian globally symmetric spaces of rank 1.)

The [[cut locus]] of any point ''p'' is equal to a hyperplane '''CP'''<sup>''n''&minus;1</sup>.  This is also the set of fixed points of the geodesic symmetry at ''p'' (less ''p'' itself).  See {{harv|Besse|1978}}.