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Difference between revisions of "Fubini-Study metric"

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  }
 
  }
 
(| x |  ^ {2}  | dx |  ^ {2} -
 
(| x |  ^ {2}  | dx |  ^ {2} -
( x, d \overline{x}\; ) ( \overline{x}\; , dx)),
+
( x, d \overline{x} ) ( \overline{x} , dx)),
 
$$
 
$$
  
 
where  $  ( \cdot , \cdot ) $
 
where  $  ( \cdot , \cdot ) $
is the scalar product in  $  \mathbf C  ^ {n+} 1 $
+
is the scalar product in  $  \mathbf C  ^ {n+ 1} $
 
and  $  | x |  ^ {2} = ( x , x ) $;  
 
and  $  | x |  ^ {2} = ( x , x ) $;  
 
the distance  $  \rho ( \widehat{x}  , \widehat{y}  ) $
 
the distance  $  \rho ( \widehat{x}  , \widehat{y}  ) $
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\frac{i}{2 \pi }
 
\frac{i}{2 \pi }
 
  }
 
  }
\partial  \overline \partial \;   \mathop{\rm ln} \  
+
\partial  \overline \partial  \mathop{\rm ln} \  
 
| z |  ^ {2} .
 
| z |  ^ {2} .
 
$$
 
$$

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

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

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=47001
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article