Fubini–Study metric

In mathematics, the Fubini–Study metric (IPA: /fubini-ʃtuːdi/) is a Kähler metric on a complex projective space CPⁿ endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.

A Hermitian form in (the vector space) Cⁿ⁺¹ defines a unitary subgroup U(n+1) in GL(n+1,C). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(n+1) action; thus it is homogeneous. Equipped with a Fubini–Study metric, CPⁿ is a symmetric space. The particular normalization on the metric depends on the application. In Riemannian geometry, one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the (2n+1)-sphere. In algebraic geometry, one uses a normalization making CPⁿ a Hodge manifold.