Fubini-Study metric

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The Hermitian metric on a complex projective space defined by the Hermitian scalar product in . It was introduced almost simultaneously by G. Fubini [1] and E. Study [2]. 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.


[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)


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.


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