# 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)