# Fubini–Study metric

# Fubini–Study metric

In mathematics, the **Fubini–Study metric** is a Kähler metric on projective Hilbert space, that is, on a complex projective space **CP***n* endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.^{[1]}^{[2]}

A Hermitian form in (the vector space) **C***n*+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, **CP***n* 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***n* a Hodge manifold.

Construction

The Fubini–Study metric arises naturally in the quotient space construction of complex projective space.

Specifically, one may define **CP***n* to be the space consisting of all complex lines in **C***n*+1, i.e., the quotient of **C***n*+1{0} by the equivalence relation relating all complex multiples of each point together. This agrees with the quotient by the diagonal group action of the multiplicative group **C*** = **C** \ {0}:

This quotient realizes **C***n*+1{0} as a complex line bundle over the base space **CP***n*. (In fact this is the so-called tautological bundle over **CP***n*.) A point of **CP***n* is thus identified with an equivalence class of (*n*+1)-tuples [*Z*0,...,*Z**n*] modulo nonzero complex rescaling; the *Z**i* are called homogeneous coordinates of the point.

`Furthermore, one may realize this quotient in two steps: since multiplication by a nonzero complex scalar`

*z*=*R**e*iθcan be uniquely thought of as the composition of a dilation by the modulus*R*followed by a counterclockwise rotation about the origin by an angle, the quotient**C***n*+1 →**CP***n*splits into two pieces.where step (a) is a quotient by the dilation **Z** ~ *R***Z** for *R* ∈ **R**+, the multiplicative group of positive real numbers, and step (b) is a quotient by the rotations **Z** ~ *e*iθ**Z**.

`The result of the quotient in (a) is the real hypersphere`

*S*2*n*+1defined by the equation |**Z**|2= |*Z*0|2 + ... + |*Z**n*|2 = 1. The quotient in (b) realizes**CP***n*=*S*2*n*+1/*S*1, where*S*1represents the group of rotations. This quotient is realized explicitly by the famousHopf fibration*S*1 →*S*2*n*+1 →**CP***n*, the fibers of which are among thegreat circlesof.As a metric quotient

`When a quotient is taken of aRiemannian manifold(ormetric spacein general), care must be taken to ensure that the quotient space is endowed with ametricthat is well-defined. For instance, if a group`

*G*acts on a Riemannian manifold (*X*,*g*), then in order for theorbit space*X*/*G*to possess an induced metric,must be constant along*G*-orbits in the sense that for any element*h*∈*G*and pair of vector fieldswe must have*g*(*Xh*,*Yh*) =*g*(*X*,*Y*).The standard Hermitian metric on **C***n*+1 is given in the standard basis by

whose realification is the standard Euclidean metric on **R**2*n*+2. This metric is *not* invariant under the diagonal action of **C***, so we are unable to directly push it down to **CP**n in the quotient. However, this metric *is* invariant under the diagonal action of *S*1 = U(1), the group of rotations. Therefore, step (b) in the above construction is possible once step (a) is accomplished.

`The`

**Fubini–Study metric**is the metric induced on the quotient**CP***n*=*S*2*n*+1/*S*1, wherecarries the so-called "round metric" endowed upon it by*restriction*of the standard Euclidean metric to the unit hypersphere.In local affine coordinates

Corresponding to a point in **CP***n* with homogeneous coordinates [*Z*0:...:*Z**n*], there is a unique set of *n* coordinates (*z*1,…,*z**n*) such that

`provided`

*Z*0 ≠ 0; specifically,*z**j*=*Z**j*/*Z*0. The (*z*1,…,*z**n*) form anaffine coordinate systemfor**CP***n*in the coordinate patch*U*0= {*Z*0 ≠ 0}. One can develop an affine coordinate system in any of the coordinate patches*U**i*= {*Z**i*≠ 0} by dividing instead by*Z**i*in the obvious manner. The*n*+1 coordinate patches*U**i*cover**CP***n*, and it is possible to give the metric explicitly in terms of the affine coordinates (*z*1,…,*z**n*) on*U**i*. The coordinate derivatives define a frameof the holomorphic tangent bundle of**CP***n*, in terms of which the Fubini–Study metric has Hermitian componentswhere |**z**|2 = |*z*1|2+...+|*z**n*|2. That is, the Hermitian matrix of the Fubini–Study metric in this frame is

`Note that each matrix element is unitary-invariant: the diagonal actionwill leave this matrix unchanged.`

Accordingly, the line element is given by

In this last expression, the summation convention is used to sum over Latin indices *i*,*j* that range from 1 to *n*.

The metric can be derived from the following Kähler potential:^{[3]}

as

Using homogeneous coordinates

An expression is also possible in the notation of homogeneous coordinates, commonly used to describe projective varieties of algebraic geometry: **Z** = [*Z*0:...:*Z**n*]. Formally, subject to suitably interpreting the expressions involved, one has

Here the summation convention is used to sum over Greek indices α β ranging from 0 to *n*, and in the last equality the standard notation for the skew part of a tensor is used:

Now, this expression for d*s*2 apparently defines a tensor on the total space of the tautological bundle **C***n*+1{0}. It is to be understood properly as a tensor on **CP***n* by pulling it back along a holomorphic section σ of the tautological bundle of **CP***n*. It remains then to verify that the value of the pullback is independent of the choice of section: this can be done by a direct calculation.

The Kähler form of this metric is

`where theare theDolbeault operators. The pullback of this is clearly independent of the choice of holomorphic section. The quantity log|`

**Z**|2is theKähler potential(sometimes called the Kähler scalar) of**CP***n*.In bra-ket coordinate notation

In quantum mechanics, the Fubini–Study metric is also known as the Bures metric.^{[4]} However, the Bures metric is typically defined in the notation of mixed states, whereas the exposition below is written in terms of a pure state. The real part of the metric is (four times) the Fisher information metric.^{[4]}

The Fubini–Study metric may be written using the bra–ket notation commonly used in quantum mechanics. To explicitly equate this notation to the homogeneous coordinates given above, let

`whereis a set oforthonormalbasis vectorsforHilbert space, theare complex numbers, andis the standard notation for a point in theprojective spaceinhomogeneous coordinates. Then, given two pointsandin the space, the distance (length of a geodesic) between them is`

or, equivalently, in projective variety notation,

`Here,is thecomplex conjugateof. The appearance ofin the denominator is a reminder thatand likewisewere not normalized to unit length; thus the normalization is made explicit here. In Hilbert space, the metric can be rather trivially interpreted as the angle between two vectors; thus it is occasionally called the`

**quantum angle**. The angle is real-valued, and runs from 0 to.`The infinitesimal form of this metric may be quickly obtained by taking, or equivalently,to obtain`

In the context of quantum mechanics, **CP**1 is called the Bloch sphere; the Fubini–Study metric is the natural metric for the geometrization of quantum mechanics. Much of the peculiar behaviour of quantum mechanics, including quantum entanglement and the Berry phase effect, can be attributed to the peculiarities of the Fubini–Study metric.

The *n* = 1 case

*n*= 1 case

`When`

*n*= 1, there is a diffeomorphismgiven bystereographic projection. This leads to the "special" Hopf fibration*S*1 →*S*3 →*S*2. When the Fubini–Study metric is written in coordinates on**CP**1, its restriction to the real tangent bundle yields an expression of the ordinary "round metric" of radius 1/2 (andGaussian curvature- on
*S*

Namely, if *z* = *x* + i*y* is the standard affine coordinate chart on the Riemann sphere **CP**1 and *x* = *r* cosθ, *y* = *r* sinθ are polar coordinates on **C**, then a routine computation shows

`whereis the round metric on the unit 2-sphere. Here φ, θ are "mathematician'sspherical coordinates" on`

*S*2coming from the stereographic projection*r*tan(φ/2) = 1, tanθ =*y*/*x*. (Many physics references interchange the roles of φ and θ.)The Kahler form is

`Choosing asvierbeinsand, the Kahler form simplifies to`

Applying the Hodge star to the Kahler form, one obtains

implying that *K* is harmonic.

The *n* = 2 case

*n*= 2 case

`The Fubini-Study metric on thecomplex projective plane`

**CP**2has been proposed as agravitational instanton, the gravitational analog of aninstanton.^{[5]}^{[3]}The metric, the connection form and the curvature are readily computed, once suitable real 4D coordinates are established. Writingfor real Cartesian coordinates, one then defines polar coordinate one-forms on the4-sphere(thequaternionic projective line) as`Theare the standard left-invariant one-form coordinate frame on the Lie group; that is, they obeyforcyclic.`

`The corresponding local affine coordinates areandthen provide`

`with the usual abbreviations thatand.`

The line element, starting with the previously given expression, is given by

The vierbeins can be immediately read off from the last expression:

That is, in the vierbein coordinate system, using roman-letter subscripts, the metric tensor is Euclidean:

`Given the vierbein, aspin connectioncan be computed; the Levi-Civita spin connection is the unique connection that istorsion-freeand covariantly constant, namely, it is the one-formthat satisfies the torsion-free condition`

and is covariantly constant, which, for spin connections, means that it is antisymmetric in the vierbein indexes:

The above is readily solved; one obtains

The curvature 2-form is defined as

and is constant:

The Ricci tensor in veirbein indexes is given by

where the curvature 2-form was expanded as a four-component tensor:

The resulting Ricci tensor is constant

so that the resulting Einstein equation

`can be solved with thecosmological constant.`

The Weyl tensor for Fubini-Study metrics in general is given by

For the *n*=2 case, the two-forms

are self-dual:

Curvature properties

`In the`

*n*= 1 special case, the Fubini–Study metric has constant sectional curvature identically equal to 4, according to the equivalence with the 2-sphere's round metric (which given a radius*R*has sectional curvature). However, for*n*> 1, the Fubini–Study metric does not have constant curvature. Its sectional curvature is instead given by the equation^{[6]}`whereis an orthonormal basis of the 2-plane σ,`

*J*:*T***CP***n*→*T***CP***n*is thecomplex structureon**CP***n*, andis the Fubini–Study metric.`A consequence of this formula is that the sectional curvature satisfiesfor all 2-planes. The maximum sectional curvature (4) is attained at aholomorphic2-plane — one for which`

*J*(σ) ⊂ σ — while the minimum sectional curvature (1) is attained at a 2-plane for which*J*(σ) is orthogonal to σ. For this reason, the Fubini–Study metric is often said to have "constant*holomorphic*sectional curvature" equal to 4.This makes **CP***n* a (non-strict) quarter pinched manifold; a celebrated theorem shows that a strictly quarter-pinched simply connected *n*-manifold must be homeomorphic to a sphere.

`The Fubini–Study metric is also anEinstein metricin that it is proportional to its ownRicci tensor: there exists a constant; such that for all`

*i*,*j*we have- .

This implies, among other things, that the Fubini–Study metric remains unchanged up to a scalar multiple under the Ricci flow. It also makes **CP***n* indispensable to the theory of general relativity, where it serves as a nontrivial solution to the vacuum Einstein field equations.

`In local affine coordinates, thecosmological constantfor`

**CP***n*is given by the dimension of the space:- .

Product metric

`The common notions of separability apply for the Fubini–Study metric. More precisely, the metric is separable on the natural product of projective spaces, theSegre embedding. That is, ifis aseparable state, so that it can be written as, then the metric is the sum of the metric on the subspaces:`

`whereandare the metrics, respectively, on the subspaces`

*A*and*B*.Connection and curvature

The fact that the metric can be derived from the Kahler potential means that the Christoffel symbols and the curvature tensors contain a lot of symmetries, and can be given a particulary simple form:^{[7]} The Christoffel symbols, in the local affine coordinates, are given by

The Riemann tensor is also particularly simple:

The Ricci tensor is

Pronunciation

A common pronunciation mistake, made especially by native English speakers, is to assume that *Study* is pronounced the same as the verb *to study*. Since it is actually a German name, the correct way to pronounce the *u* in *Study* is the same as the *u* in *Fubini*. In terms of phonetics: ʃtuːdi.

See also

Non-linear sigma model

Kaluza–Klein theory

Arakelov height

## References

*Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti*,

**63**pp. 502–513

*Math. Ann.*,

**60**pp. 321–378

*Physics Reports*

**66**(1980) pp 213-393.

*Physics Letters*

**A 374**pp. 4801. doi:10.1016/j.physleta.2010.10.005

*Phys. Rev. Lett.*

**37.**pp.1251—1254

*Riemannian Geometry*, Translations of Mathematical Monographs No. 149 (1995), American Mathematics Society.