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The [[Hermitian metric|Hermitian metric]] on a complex projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f0418601.png" /> defined by the Hermitian scalar product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f0418602.png" />. It was introduced almost simultaneously by G. Fubini [[#References|[1]]] and E. Study [[#References|[2]]]. The Fubini–Study metric is given by the formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f0418603.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f0418604.png" /> is the scalar product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f0418605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f0418606.png" />; the distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f0418607.png" /> between the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f0418608.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f0418609.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f04186010.png" />, is determined from the formula
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The [[Hermitian metric|Hermitian metric]] on a complex projective space  $  \mathbf C P  ^ {n} $
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defined by the Hermitian scalar product in $  \mathbf C ^ {n + 1 } $.  
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It was introduced almost simultaneously by G. Fubini [[#References|[1]]] and E. Study [[#References|[2]]]. The Fubini–Study metric is given by the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f04186011.png" /></td> </tr></table>
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$$
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ds  ^ {2}  = \
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{
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\frac{1}{| x |  ^ {4} }
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}
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(| x |  ^ {2}  | dx |  ^ {2} -
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( x, d \overline{x} ) ( \overline{x} , dx)),
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$$
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where  $  ( \cdot , \cdot ) $
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is the scalar product in  $  \mathbf C  ^ {n+ 1} $
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and  $  | x |  ^ {2} = ( x , x ) $;  
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the distance  $  \rho ( \widehat{x}  , \widehat{y}  ) $
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between the points  $  \widehat{x}  = \mathbf C x $,
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$  y = \mathbf C y $,
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where  $  x, y \in \mathbf C ^ {n + 1 } \setminus  \{ 0 \} $,
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is determined from the formula
 +
 
 +
$$
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\cos  \rho ( \widehat{x}  , \widehat{y}  )  = \
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 +
\frac{| ( x, y) | }{| x | \cdot | y | }
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.
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$$
  
 
The Fubini–Study metric is Kählerian (and is even a Hodge metric); its associated Kähler form is
 
The Fubini–Study metric is Kählerian (and is even a Hodge metric); its associated Kähler form is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f04186012.png" /></td> </tr></table>
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$$
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\omega  = {
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\frac{i}{2 \pi }
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}
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\partial  \overline \partial  \ln | z |  ^ {2} .
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$$
  
The Fubini–Study metric is, up to proportionality, the unique Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f04186013.png" /> that is invariant under the unitary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f04186014.png" />, which preserves the scalar product. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041860/f04186015.png" />, endowed with the Fubini–Study metric, is a compact Hermitian symmetric space of rank 1. It is also called an elliptic Hermitian space.
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The Fubini–Study metric is, up to proportionality, the unique Riemannian metric on $  \mathbf C P  ^ {n} $
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that is invariant under the unitary group $  U ( n + 1) $,  
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which preserves the scalar product. The space $  \mathbf C P  ^ {n} $,  
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endowed with the Fubini–Study metric, is a compact Hermitian symmetric space of rank 1. It is also called an elliptic Hermitian space.
  
 
====References====
 
====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 géometrie 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></table>
 
<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 géometrie 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></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 07:51, 13 May 2022


The 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 [1] and E. Study [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.

References

[1] G. Fubini, "Sulle metriche definite da una forme Hermitiana" Atti Istit. Veneto , 63 (1904) pp. 502–513
[2] E. Study, "Kürzeste Wege im komplexen Gebiet" Math. Ann. , 60 (1905) pp. 321–378
[3] E. Cartan, "Leçons sur la géometrie projective complexe" , Gauthier-Villars (1950)
[4] S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962)
[5] S.S. Chern, "Complex manifolds" , Univ. Recife (1959)

Comments

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

For Hodge and Kähler metrics cf. Kähler metric.

References

[a1] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978)
[a2] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
[a3] E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian)
How to Cite This Entry:
Fubini-Study metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fubini-Study_metric&oldid=16968
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article