Difference between revisions of "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==== | ====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. [[ | + | For Hodge and Kähler metrics cf. [[Kähler metric]]. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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=22475
Fubini-Study metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fubini-Study_metric&oldid=22475
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article