View source for Fubini-Study metric

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The [[Hermitian metric|Hermitian metric]] on a complex projective space  $  \mathbf C P  ^ {n} $
defined by the Hermitian scalar product in  $  \mathbf C ^ {n + 1 } $. 
It was introduced almost simultaneously by G. Fubini [[#References|[1]]] and E. Study [[#References|[2]]]. The Fubini–Study metric is given by the formula

$$ 
ds  ^ {2}  = \ 
{
\frac{1}{| x |  ^ {4} }
 }
(| x |  ^ {2}  | dx |  ^ {2} -
( x, d \overline{x} ) ( \overline{x} , dx)),
$$

where  $  ( \cdot , \cdot ) $
is the scalar product in  $  \mathbf C  ^ {n+ 1} $
and  $  | x |  ^ {2} = ( x , x ) $; 
the distance  $  \rho ( \widehat{x}  , \widehat{y}  ) $
between the points  $  \widehat{x}  = \mathbf C x $, 
$  y = \mathbf C y $, 
where  $  x, y \in \mathbf C ^ {n + 1 } \setminus  \{ 0 \} $, 
is determined from the formula

$$ 
\cos  \rho ( \widehat{x}  , \widehat{y}  )  = \ 

\frac{| ( x, y) | }{| x | \cdot | y | }
 .
$$

The Fubini–Study metric is Kählerian (and is even a Hodge metric); its associated Kähler form is

$$ 
\omega  =  {
\frac{i}{2 \pi }
 }
\partial  \overline \partial   \ln | z |  ^ {2} .
$$

The Fubini–Study metric is, up to proportionality, the unique Riemannian metric on  $  \mathbf C P  ^ {n} $
that is invariant under the unitary group  $  U ( n + 1) $, 
which preserves the scalar product. The space  $  \mathbf C P  ^ {n} $, 
endowed with the Fubini–Study metric, is a compact Hermitian symmetric space of rank 1. It is also called an elliptic Hermitian space.

====Comments====
Reference [[#References|[a1]]] below is an extended and revised version of [[#References|[4]]]. The Fubini–Study metric is extensively used in (multi-dimensional) complex analysis, [[#References|[a2]]], [[#References|[a3]]].

For Hodge and Kähler metrics cf. [[Kähler metric]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> G. Fubini, "Sulle metriche definite da una forme Hermitiana"  ''Atti Istit. Veneto'' , '''63'''  (1904)  pp. 502–513</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> E. Study, "Kürzeste Wege im komplexen Gebiet"  ''Math. Ann.'' , '''60'''  (1905)  pp. 321–378</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> E. Cartan, "Leçons sur la géométrie projective complexe" , Gauthier-Villars  (1950)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press  (1962)</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top"> S.S. Chern, "Complex manifolds" , Univ. Recife  (1959)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top"> E.M. Chirka, "Complex analytic sets" , Kluwer  (1989)  (Translated from Russian)</TD></TR>
</table>