Difference between revisions of "Fubini-Study metric"
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} | } | ||
(| x | ^ {2} | dx | ^ {2} - | (| x | ^ {2} | dx | ^ {2} - | ||
− | ( x, d \overline{x} | + | ( x, d \overline{x} ) ( \overline{x} , dx)), |
$$ | $$ | ||
where $ ( \cdot , \cdot ) $ | where $ ( \cdot , \cdot ) $ | ||
− | is the scalar product in $ \mathbf C ^ {n+} | + | 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} ) $ | ||
Line 46: | Line 46: | ||
\frac{i}{2 \pi } | \frac{i}{2 \pi } | ||
} | } | ||
− | \partial \overline \partial | + | \partial \overline \partial \ln | z | ^ {2} . |
− | | z | ^ {2} . | ||
$$ | $$ | ||
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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. | ||
− | |||
− | |||
− | |||
====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"> 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 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> | ||
+ | <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> |
Revision as of 06:11, 13 August 2023
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 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) |
[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) |
Fubini-Study metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fubini-Study_metric&oldid=47001