Construction Fubini–Study metric

1 construction

1.1 metric quotient
1.2 in local affine coordinates
1.3 homogeneous coordinates
1.4 n = 1 case

construction

the fubini–study metric arises naturally in quotient space construction of complex projective space.

specifically, 1 may define cp space consisting of complex lines in c, i.e., quotient of c\{0} equivalence relation relating complex multiples of each point together. agrees quotient diagonal group action of multiplicative group c = c \ {0}:

c
p


n


=

{

z

=
[

z

0


,

z

1


,
…
,

z

n


]
∈



c



n
+
1


∖
{
0
}

}


/

{

z

∼
c

z

,
c
∈


c


∗


}
.


{\displaystyle \mathbf {cp} ^{n}=\left\{\mathbf {z} =[z_{0},z_{1},\ldots ,z_{n}]\in {\mathbf {c} }^{n+1}\setminus \{0\}\,\right\}/\{\mathbf {z} \sim c\mathbf {z} ,c\in \mathbf {c} ^{*}\}.}

this quotient realizes c\{0} complex line bundle on base space cp. (in fact so-called tautological bundle on cp.) point of cp identified equivalence class of (n+1)-tuples [z0,...,zn] modulo nonzero complex rescaling; zi called homogeneous coordinates of point.

furthermore, 1 may realize quotient in 2 steps: since multiplication nonzero complex scalar z = r e can uniquely thought of composition of dilation modulus r followed counterclockwise rotation origin angle



θ


{\displaystyle \theta }

, quotient c → cp splits 2 pieces.

c


n
+
1


∖
{
0
}




⟶


(
a
)





s

2
n
+
1






⟶


(
b
)






c
p


n




{\displaystyle \mathbf {c} ^{n+1}\setminus \{0\}{\stackrel {(a)}{\longrightarrow }}s^{2n+1}{\stackrel {(b)}{\longrightarrow }}\mathbf {cp} ^{n}}

where step (a) quotient dilation z ~ rz r ∈ r, multiplicative group of positive real numbers, , step (b) quotient rotations z ~ ez.

the result of quotient in (a) real hypersphere s defined equation |z| = |z0| + ... + |zn| = 1. quotient in (b) realizes cp = s/s, s represents group of rotations. quotient realized explicitly famous hopf fibration s → s → cp, fibers of among great circles of




s

2
n
+
1




{\displaystyle s^{2n+1}}

.

as metric quotient

when quotient taken of riemannian manifold (or metric space in general), care must taken ensure quotient space endowed metric well-defined. instance, if group g acts on riemannian manifold (x,g), in order orbit space x/g possess induced metric,



g


{\displaystyle g}

must constant along g-orbits in sense element h ∈ g , pair of vector fields



x
,
y


{\displaystyle x,y}

must have g(xh,yh) = g(x,y).

the standard hermitian metric on c given in standard basis by

d

s

2


=
d

z

⊗
d



z

¯


=
d

z

0


⊗
d



z

0


¯


+
⋯
+
d

z

n


⊗
d



z

n


¯




{\displaystyle ds^{2}=d\mathbf {z} \otimes d{\overline {\mathbf {z} }}=dz_{0}\otimes d{\overline {z_{0}}}+\cdots +dz_{n}\otimes d{\overline {z_{n}}}}

whose realification standard euclidean metric on r. metric not invariant under diagonal action of c, unable directly push down cp in quotient. however, metric invariant under diagonal action of s = u(1), group of rotations. therefore, step (b) in above construction possible once step (a) accomplished.

the fubini–study metric metric induced on quotient cp = s/s,




s

2
n
+
1




{\displaystyle s^{2n+1}}

carries so-called round metric endowed upon restriction of standard euclidean metric unit hypersphere.

in local affine coordinates

corresponding point in cp homogeneous coordinates (z0,...,zn), there unique set of n coordinates (z1,…,zn) such that

[

z

0


,
…
,

z

n


]

∼

[
1
,

z

1


,
…
,

z

n


]
,


{\displaystyle [z_{0},\dots ,z_{n}]{\sim }[1,z_{1},\dots ,z_{n}],}

provided z0 ≠ 0; specifically, zj = zj/z0. (z1,…,zn) form affine coordinate system cp in coordinate patch u0 = {z0 ≠ 0}. 1 can develop affine coordinate system in of coordinate patches ui = {zi ≠ 0} dividing instead zi in obvious manner. n+1 coordinate patches ui cover cp, , possible give metric explicitly in terms of affine coordinates (z1,…,zn) on ui. coordinate derivatives define frame



{

∂

1


,
…
,

∂

n


}


{\displaystyle \{\partial _{1},\ldots ,\partial _{n}\}}

of holomorphic tangent bundle of cp, in terms of fubini–study metric has hermitian components

h

i



j
¯





=
h
(

∂

i


,




∂
¯




j


)
=



(
1
+

|


z



|


2


)

δ

i



j
¯





−




z
¯




i



z

j




(
1
+

|


z



|


2



)

2





.


{\displaystyle h_{i{\bar {j}}}=h(\partial _{i},{\bar {\partial }}_{j})={\frac {(1+|\mathbf {z} |^{2})\delta _{i{\bar {j}}}-{\bar {z}}_{i}z_{j}}{(1+|\mathbf {z} |^{2})^{2}}}.}

where |z| = |z1|+...+|zn|. is, hermitian matrix of fubini–study metric in frame is

(



h

i



j
¯







)


=


1

(
1
+

|


z



|


2



)

2






[




1
+

|


z



|


2


−

|


z

1




|


2




−




z
¯




1



z

2




⋯


−




z
¯




1



z

n






−




z
¯




2



z

1




1
+

|


z



|


2


−

|


z

2




|


2




⋯


−




z
¯




2



z

n






⋮


⋮


⋱


⋮




−




z
¯




n



z

1




−




z
¯




n



z

2




⋯


1
+

|


z



|


2


−

|


z

n




|


2






]



{\displaystyle {\bigl (}h_{i{\bar {j}}}{\bigr )}={\frac {1}{(1+|\mathbf {z} |^{2})^{2}}}\left[{\begin{array}{cccc}1+|\mathbf {z} |^{2}-|z_{1}|^{2}&-{\bar {z}}_{1}z_{2}&\cdots &-{\bar {z}}_{1}z_{n}\\-{\bar {z}}_{2}z_{1}&1+|\mathbf {z} |^{2}-|z_{2}|^{2}&\cdots &-{\bar {z}}_{2}z_{n}\\\vdots &\vdots &\ddots &\vdots \\-{\bar {z}}_{n}z_{1}&-{\bar {z}}_{n}z_{2}&\cdots &1+|\mathbf {z} |^{2}-|z_{n}|^{2}\end{array}}\right]}

note each matrix element unitary-invariant: diagonal action




z

↦

e

i
θ



z



{\displaystyle \mathbf {z} \mapsto e^{i\theta }\mathbf {z} }

leave matrix unchanged.

accordingly, line element given by

d

s

2





=



(
1
+

|


z



|


2


)

|

d

z



|


2


−
(




z

¯



⋅
d

z

)
(

z

⋅
d




z

¯



)


(
1
+

|


z



|


2



)

2











=



(
1
+

z

i






z
¯




i


)
d

z

j


d




z
¯




j


−




z
¯




j



z

i


d

z

j


d




z
¯




i




(
1
+

z

i






z
¯




i



)

2





.






{\displaystyle {\begin{aligned}ds^{2}&={\frac {(1+|\mathbf {z} |^{2})|d\mathbf {z} |^{2}-({\bar {\mathbf {z} }}\cdot d\mathbf {z} )(\mathbf {z} \cdot d{\bar {\mathbf {z} }})}{(1+|\mathbf {z} |^{2})^{2}}}\\&={\frac {(1+z_{i}{\bar {z}}^{i})dz_{j}d{\bar {z}}^{j}-{\bar {z}}^{j}z_{i}dz_{j}d{\bar {z}}^{i}}{(1+z_{i}{\bar {z}}^{i})^{2}}}.\end{aligned}}}

in last expression, summation convention used sum on latin indices i,j range 1 n.

the metric can derived following kähler potential:

k
=
ln
⁡
(
1
+

δ

i

j

∗





z

i






z
¯





j

∗




)


{\displaystyle k=\ln(1+\delta _{ij^{*}}z^{i}{\bar {z}}^{j^{*}})}

as

g

i

j

∗




=

k

i

j

∗




=



∂

2



∂

z

i


∂




z
¯





j

∗







k


{\displaystyle g_{ij^{*}}=k_{ij^{*}}={\frac {\partial ^{2}}{\partial z^{i}\partial {\bar {z}}^{j^{*}}}}k}



homogeneous coordinates

an expression possible in homogeneous coordinates z = [z0,...,zn]. formally, subject suitably interpreting expressions involved, 1 has

d

s

2





=




|


z



|


2



|

d

z



|


2


−
(




z

¯



⋅
d

z

)
(

z

⋅
d




z

¯



)



|


z



|


4











=




z

α






z
¯




α


d

z

β


d




z
¯




β


−




z
¯




α



z

β


d

z

α


d




z
¯




β




(

z

α






z
¯




α



)

2











=



2

z

[
α


d

z

β
]





z
¯



[
α






d
z

¯



β
]





(

z

α





z
¯



α


)


2




.






{\displaystyle {\begin{aligned}ds^{2}&={\frac {|\mathbf {z} |^{2}|d\mathbf {z} |^{2}-({\bar {\mathbf {z} }}\cdot d\mathbf {z} )(\mathbf {z} \cdot d{\bar {\mathbf {z} }})}{|\mathbf {z} |^{4}}}\\&={\frac {z_{\alpha }{\bar {z}}^{\alpha }dz_{\beta }d{\bar {z}}^{\beta }-{\bar {z}}^{\alpha }z_{\beta }dz_{\alpha }d{\bar {z}}^{\beta }}{(z_{\alpha }{\bar {z}}^{\alpha })^{2}}}\\&={\frac {2z_{[\alpha }dz_{\beta ]}{\overline {z}}^{[\alpha }{\overline {dz}}^{\beta ]}}{\left(z_{\alpha }{\overline {z}}^{\alpha }\right)^{2}}}.\end{aligned}}}

here summation convention used sum on greek indices α β ranging 0 n, , in last equality standard notation skew part of tensor used:

z

[
α



w

β
]


=


1
2



(

z

α



w

β


−

z

β



w

α


)

.


{\displaystyle z_{[\alpha }w_{\beta ]}={\frac {1}{2}}\left(z_{\alpha }w_{\beta }-z_{\beta }w_{\alpha }\right).}

now, expression ds apparently defines tensor on total space of tautological bundle c\{0}. understood tensor on cp pulling along holomorphic section σ of tautological bundle of cp. remains verify value of pullback independent of choice of section: can done direct calculation.

the kähler form of metric is, overall constant normalization,

ω
=
i
∂


∂
¯


log
⁡

|


z



|


2




{\displaystyle \omega =i\partial {\overline {\partial }}\log |\mathbf {z} |^{2}}

the pullback of independent of choice of holomorphic section. quantity log|z| kähler scalar of cp.

the n = 1 case

when n = 1, there diffeomorphism




s

2


≅


c
p


1




{\displaystyle s^{2}\cong \mathbb {cp} ^{1}}

given stereographic projection. leads special hopf fibration s → s → s. when fubini–study metric written in coordinates on cp, restriction real tangent bundle yields expression of ordinary round metric of radius 1/2 (and gaussian curvature 4) on s.

namely, if z = x + iy standard affine coordinate chart on riemann sphere cp , x = r cosθ, y = r sinθ polar coordinates on c, routine computation shows

d

s

2


=



re
⁡
(
d
z
⊗
d


z
¯


)



(
1
+

|

z


|


2


)


2




=



d

x

2


+
d

y

2





(
1
+

r

2


)


2




=


1
4


(
d

ϕ

2


+

sin

2


⁡
ϕ

d

θ

2


)
=


1
4


d

s

u
s


2




{\displaystyle ds^{2}={\frac {\operatorname {re} (dz\otimes d{\overline {z}})}{\left(1+|z|^{2}\right)^{2}}}={\frac {dx^{2}+dy^{2}}{\left(1+r^{2}\right)^{2}}}={\frac {1}{4}}(d\phi ^{2}+\sin ^{2}\phi \,d\theta ^{2})={\frac {1}{4}}ds_{us}^{2}}

where



d

s

u
s


2




{\displaystyle ds_{us}^{2}}

round metric on unit 2-sphere. here φ, θ mathematician s spherical coordinates on s coming stereographic projection r tan(φ/2) = 1, tanθ = y/x. (many physics references interchange roles of φ , θ.)