A Hermitian form in (the vector space) C 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, CP 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 (2 n +1)-sphere. In algebraic geometry, one uses a normalization making CP a Hodge manifold.