Fubini-Study metric

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)