The Hermitian metric on a complex projective space defined by the Hermitian scalar product in . It was introduced almost simultaneously by G. Fubini  and E. Study . The Fubini–Study metric is given by the formula
where is the scalar product in and ; the distance between the points , , where , is determined from the formula
The Fubini–Study metric is Kählerian (and is even a Hodge metric); its associated Kähler form is
The Fubini–Study metric is, up to proportionality, the unique Riemannian metric on that is invariant under the unitary group , which preserves the scalar product. The space , endowed with the Fubini–Study metric, is a compact Hermitian symmetric space of rank 1. It is also called an elliptic Hermitian space.
|||G. Fubini, "Sulle metriche definite da una forme Hermitiana" Atti Istit. Veneto , 63 (1904) pp. 502–513|
|||E. Study, "Kürzeste Wege im komplexen Gebiet" Math. Ann. , 60 (1905) pp. 321–378|
|||E. Cartan, "Leçons sur la géometrie projective complexe" , Gauthier-Villars (1950)|
|||S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962)|
|||S.S. Chern, "Complex manifolds" , Univ. Recife (1959)|
Reference [a1] below is an extended and revised version of . The Fubini–Study metric is extensively used in (multi-dimensional) complex analysis, [a2], [a3].
For Hodge and Kähler metrics cf. Kähler metric.
|[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=22475