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Difference between revisions of "Fubini-Study metric"

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which preserves the scalar product. The space  $  \mathbf C P  ^ {n} $,  
 
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.
 
endowed with the Fubini–Study metric, is a compact Hermitian symmetric space of rank 1. It is also called an elliptic Hermitian space.
 
====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>
 
  
 
====Comments====
 
====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]]].
 
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|Kähler metric]].
+
For Hodge and Kähler metrics cf. [[Kähler metric]].
  
 
====References====
 
====References====
<table><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>
+
<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>

Latest revision as of 16:10, 18 July 2024


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.

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

[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 géométrie 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)
[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=52362
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article