Editing Complex projective space (section)

==Differential geometry==
The natural metric on '''CP'''<sup>''n''</sup> is the [[Fubini–Study metric]], and its holomorphic isometry group is the [[projective unitary group]] PU(''n''+1), where the stabilizer of a point is

:<math>\mathrm{P}(1\times \mathrm{U}(n)) \cong \mathrm{PU}(n).</math>

It is a [[Hermitian symmetric space]] {{harv|Kobayashi|Nomizu|1996}}, represented as a coset space

:<math>U(n+1)/(U(1) \times U(n)) \cong SU(n+1)/S(U(1) \times U(n)).</math>

The geodesic symmetry at a point ''p'' is the unitary transformation that fixes ''p'' and is the negative identity on the orthogonal complement of the line represented by ''p''.

===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}}.

===Sectional curvature pinching===
It has [[sectional curvature]] ranging from 1/4 to 1, and is the roundest manifold that is not a sphere (or covered by a sphere): by the [[Riemannian geometry#Pinched sectional curvature|1/4-pinched sphere theorem]], any complete, simply connected [[Riemannian manifold]] with curvature strictly between 1/4 and 1 is diffeomorphic to the sphere. Complex projective space shows that 1/4 is sharp.  Conversely, if a complete simply connected Riemannian manifold has sectional curvatures in the closed interval [1/4,1], then it is either diffeomorphic to the sphere, or isometric to the complex projective space, the [[quaternionic projective space]], or else the [[Cayley plane]] F<sub>4</sub>/Spin(9); see {{harv|Brendle|Schoen|2008}}.

===Spin structure===
The odd-dimensional projective spaces can be given a [[spin structure]], the even-dimensional ones cannot.