Inmathematics, theFubini–Study metric (IPA: /fubini-ʃtuːdi/) is aKähler metric on acomplex projective spaceCPⁿ endowed with aHermitian form. Thismetric was originally described in 1904 and 1905 byGuido Fubini andEduard Study.^[1][2]

AHermitian form in (the vector space)Cⁿ⁺¹ defines aunitary subgroup U(n+1) in GL(n+1,C). A Fubini–Study metric is determined up tohomothety (overall scaling) by invariance under such a U(n+1) action; thus it ishomogeneous. Equipped with a Fubini–Study metric,CPⁿ is asymmetric space. The particular normalization on the metric depends on the application. InRiemannian geometry, one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the(2n+1)-sphere. Inalgebraic geometry, one uses a normalization makingCPⁿ aHodge manifold.

The Fubini–Study metric arises naturally in thequotient space construction ofcomplex projective space.

Specifically, one may defineCPⁿ to be the space consisting of all complex lines inCⁿ⁺¹, i.e., the quotient ofCⁿ⁺¹\{0} by theequivalence relation relating all complex multiples of each point together. This agrees with the quotient by the diagonalgroup action of the multiplicative groupC^* = C \ {0}:

${\displaystyle {\mathbf{C}\mathbf{P}}^n=\left\{\mathbf{Z}=[Z_0,Z_1,\dots ,Z_n]\in {\mathbf{C}}^{n+1}\setminus \{0\}\right\}/\{\mathbf{Z}\sim c\mathbf{Z},c\in {\mathbf{C}}^{\ast }\}.}$

This quotient realizesCⁿ⁺¹\{0} as a complexline bundle over the base spaceCPⁿ. (In fact this is the so-calledtautological bundle overCPⁿ.) A point ofCPⁿ is thus identified with an equivalence class of (n+1)-tuples [Z₀,...,Z_n] modulo nonzero complex rescaling; theZ_i are calledhomogeneous coordinates of the point.

Furthermore, one may realize this quotient mapping in two steps: since multiplication by a nonzero complex scalarz = R e^iθ can be uniquely thought of as the composition of a dilation by the modulusR followed by a counterclockwise rotation about the origin by an angle${\displaystyle \theta }$, the quotient mappingCⁿ⁺¹\{0} → CPⁿ splits into two pieces,

${\displaystyle {\mathbf{C}}^{n+1}\setminus \{0\}\stackrel{(a)}{⟶}{S}^{2n+1}\stackrel{(b)}{⟶}{\mathbf{C}\mathbf{P}}^n}$

where step (a) is a quotient by the dilationZ ~ RZ forR ∈ R⁺, the multiplicative group ofpositive real numbers, and step (b) is a quotient by the rotationsZ ~ e^iθZ.

The result of the quotient in (a) is the real hypersphereS²ⁿ⁺¹ defined by the equation |Z|² = |Z₀|² + ... + |Z_n|² = 1. The quotient in (b) realizesCPⁿ = S²ⁿ⁺¹/S¹, whereS¹ represents the group of rotations. This quotient is realized explicitly by the famousHopf fibrationS¹ → S²ⁿ⁺¹ → CPⁿ, the fibers of which are among thegreat circles of${\displaystyle S^{2n+1}}$.

When a quotient is taken of aRiemannian manifold (ormetric space in general), care must be taken to ensure that the quotient space is endowed with ametric that is well-defined. For instance, if a groupG acts on a Riemannian manifold (X,g), then in order for theorbit spaceX/G to possess an induced metric,${\displaystyle g}$ must be constant alongG-orbits in the sense that for any elementh ∈ G and pair of vector fields${\displaystyle X,Y}$ we must haveg(Xh,Yh) = g(X,Y).

The standardHermitian metric onCⁿ⁺¹ is given in the standard basis by

${\displaystyle d{s}^2=d\mathbf{Z}\otimes d\overline{\mathbf{Z}}=d{Z}_0\otimes d{\overline{Z}}_0+\cdots +d{Z}_n\otimes d{\overline{Z}}_n}$

whose realification is the standardEuclidean metric onR²ⁿ⁺². This metric isnot invariant under the diagonal action ofC^*, so we are unable to directly push it down toCPⁿ in the quotient. However, this metricis invariant under the diagonal action ofS¹ = U(1), the group of rotations. Therefore, step (b) in the above construction is possible once step (a) is accomplished.

TheFubini–Study metric is the metric induced on the quotientCPⁿ = S²ⁿ⁺¹/S¹, where${\displaystyle S^{2n+1}}$ carries the so-called "round metric" endowed upon it byrestriction of the standard Euclidean metric to the unit hypersphere.

Corresponding to a point inCPⁿ with homogeneous coordinates${\displaystyle [Z_0:\cdots :Z_n]}$, there is a unique set ofn coordinates${\displaystyle (z_1,\dots ,z_n)}$ such that

${\displaystyle [Z_0:\cdots :Z_n]\sim [1,z_1,\dots ,z_n],}$

provided${\displaystyle Z_0\ne 0}$; specifically,${\displaystyle z_j=Z_j/Z_0}$. The${\displaystyle (z_1,\dots ,z_n)}$ form anaffine coordinate system forCPⁿ in the coordinate patch${\displaystyle U_0=\{Z_0\ne 0\}}$. One can develop an affine coordinate system in any of the coordinate patches${\displaystyle U_i=\{Z_i\ne 0\}}$ by dividing instead by${\displaystyle Z_i}$ in the obvious manner. Then+1 coordinate patches${\displaystyle U_i}$ coverCPⁿ, and it is possible to give the metric explicitly in terms of the affine coordinates${\displaystyle (z_1,\dots ,z_n)}$ on${\displaystyle U_i}$. The coordinate derivatives define a frame${\displaystyle \{{\mathrm{\partial }}_1,\dots ,{\mathrm{\partial }}_n\}}$ of the holomorphic tangent bundle ofCPⁿ, in terms of which the Fubini–Study metric has Hermitian components

${\displaystyle g_{i\overline{j}}=h({\mathrm{\partial }}_i,{\overline{\mathrm{\partial }}}_j)=\frac{\left(1+|\mathbf{z}|{\phantom{l}}^2\right){\delta }_{i\overline{j}}-{\overline{z}}_i{z}_j}{{\left(1+|\mathbf{z}|{\phantom{l}}^2\right)}^2}.}$

where |z|² = |z₁|² + ... + |z_n|². That is, theHermitian matrix of the Fubini–Study metric in this frame is

Note that each matrix element is unitary-invariant: the diagonal action${\displaystyle \mathbf{z}\mapsto e^{i\theta }\mathbf{z}}$ will leave this matrix unchanged.

Accordingly, the line element is given by

In this last expression, thesummation convention is used to sum over Latin indicesi,j that range from 1 to n.

The metric can be derived from the followingKähler potential:^[3]

${\displaystyle K=\ln (1+z_i{\overline{z}}^i)=\ln (1+{\delta }_{i\overline{j}}{z}^i{\overline{z}}^j)}$

as

${\displaystyle g_{i\overline{j}}=K_{i\overline{j}}=\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^i\mathrm{\partial }{\overline{z}}^j}K}$

An expression is also possible in the notation ofhomogeneous coordinates, commonly used to describeprojective varieties ofalgebraic geometry:Z = [Z₀:...: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 ton, and in the last equality the standard notation for the skew part of a tensor is used:

${\displaystyle Z_{[\alpha }{W}_{\beta ]}=\frac{1}{2}\left(Z_{\alpha }{W}_{\beta }-Z_{\beta }{W}_{\alpha }\right).}$

Now, this expression for ds² apparently defines a tensor on the total space of the tautological bundleCⁿ⁺¹\{0}. It is to be understood properly as a tensor onCPⁿ by pulling it back along a holomorphic section σ of the tautological bundle ofCPⁿ. 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.

TheKähler form of this metric is

${\displaystyle \omega =\frac{i}{2}\mathrm{\partial }\overline{\mathrm{\partial }}\log |\mathbf{Z}{|}^2}$

where the${\displaystyle \mathrm{\partial },\overline{\mathrm{\partial }}}$ are theDolbeault operators. The pullback of this is clearly independent of the choice of holomorphic section. The quantity log|Z|² is theKähler potential (sometimes called the Kähler scalar) ofCPⁿ.

Inquantum mechanics, the Fubini–Study metric is also known as theBures metric.^[4] However, the Bures metric is typically defined in the notation ofmixed states, whereas the exposition below is written in terms of apure state. The real part of the metric is (a quarter of) theFisher information metric.^[4]

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

${\displaystyle |\psi ⟩=\sum _{k=0}^n{Z}_k|e_k⟩=[Z_0:Z_1:\dots :Z_n]}$

where${\displaystyle \{|e_k⟩\}}$ is a set oforthonormalbasis vectors forHilbert space, the${\displaystyle Z_k}$ are complex numbers, and${\displaystyle Z_{\alpha }=[Z_0:Z_1:\dots :Z_n]}$ is the standard notation for a point in theprojective spaceCPⁿ inhomogeneous coordinates. Then, given two points${\displaystyle |\psi ⟩=Z_{\alpha }}$ and${\displaystyle |\phi ⟩=W_{\alpha }}$ in the space, the distance (length of a geodesic) between them is

${\displaystyle \gamma (\psi ,\phi )=\arccos \sqrt{\frac{⟨\psi |\phi ⟩⟨\phi |\psi ⟩}{⟨\psi |\psi ⟩⟨\phi |\phi ⟩}}}$

or, equivalently, in projective variety notation,

${\displaystyle \gamma (\psi ,\phi )=\gamma (Z,W)=\arccos \sqrt{\frac{Z_{\alpha }{\overline{W}}^{\alpha }{W}_{\beta }{\overline{Z}}^{\beta }}{Z_{\alpha }{\overline{Z}}^{\alpha }{W}_{\beta }{\overline{W}}^{\beta }}}.}$

Here,${\displaystyle {\overline{Z}}^{\alpha }}$ is thecomplex conjugate of${\displaystyle Z_{\alpha }}$. The appearance of${\displaystyle ⟨\psi |\psi ⟩}$ in the denominator is a reminder that${\displaystyle |\psi ⟩}$ and likewise${\displaystyle |\phi ⟩}$ were not normalized to unit length; thus the normalization is made explicit here. In Hilbert space, the metric can be interpreted as the angle between two vectors; thus it is occasionally called thequantum angle. The angle is real-valued, and runs from 0 to${\displaystyle \pi /2}$.

The infinitesimal form of this metric may be quickly obtained by taking${\displaystyle \phi =\psi +\delta \psi }$, or equivalently,${\displaystyle W_{\alpha }=Z_{\alpha }+d{Z}_{\alpha }}$ to obtain

${\displaystyle d{s}^2=\frac{⟨\delta \psi |\delta \psi ⟩}{⟨\psi |\psi ⟩}-\frac{⟨\delta \psi |\psi ⟩⟨\psi |\delta \psi ⟩}{{⟨\psi |\psi ⟩}^2}.}$

In the context ofquantum mechanics,CP¹ is called theBloch sphere; the Fubini–Study metric is the naturalmetric for the geometrization of quantum mechanics. Much of the peculiar behaviour of quantum mechanics, includingquantum entanglement and theBerry phase effect, can be attributed to the peculiarities of the Fubini–Study metric.

Whenn = 1, there is a diffeomorphism${\displaystyle S^2\cong {\mathbf{C}\mathbf{P}}^1}$ given bystereographic projection. This leads to the "special" Hopf fibrationS¹ → S³ → S². When the Fubini–Study metric is written in coordinates onCP¹, its restriction to the real tangent bundle yields an expression of the ordinary "round metric" of radius 1/2 (andGaussian curvature 4) onS².

Namely, ifz = x + iy is the standard affine coordinate chart on theRiemann sphereCP¹ andx = r cos θ,y = r sin θ are polar coordinates onC, then a routine computation shows

${\displaystyle d{s}^2=\frac{\mathrm{Re}(dz\otimes d\overline{z})}{{\left(1+|\mathbf{z}|{\phantom{l}}^2\right)}^2}=\frac{d{x}^2+d{y}^2}{{\left(1+r^2\right)}^2}=\frac{1}{4}(d{\phi }^2+{\sin }^2\phi d{\theta }^2)=\frac{1}{4}d{s}_{us}^2}$

where${\displaystyle d{s}_{us}^2}$ is the round metric on the unit 2-sphere. Here φ, θ are "mathematician'sspherical coordinates" onS² coming from the stereographic projectionr tan(φ/2) = 1, tan θ = y/x. (Many physics references interchange the roles of φ and θ.)

TheKähler form is

${\displaystyle K=\frac{i}{2}\frac{dz\wedge d\overline{z}}{{\left(1+z\overline{z}\right)}^2}=\frac{dx\wedge dy}{{\left(1+x^2+y^2\right)}^2}}$

Choosing asvierbeins${\displaystyle e^1=dx/(1+r^2)}$ and${\displaystyle e^2=dy/(1+r^2)}$, the Kähler form simplifies to

${\displaystyle K=e^1\wedge e^2}$

Applying theHodge star to the Kähler form, one obtains

${\displaystyle \ast K=1}$

implying thatK isharmonic.

The Fubini–Study metric on thecomplex projective planeCP² has 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. Writing${\displaystyle (x,y,z,t)}$ for real Cartesian coordinates, one then defines polar coordinate one-forms on the4-sphere (thequaternionic projective line) as

The${\displaystyle {\sigma }_1,{\sigma }_2,{\sigma }_3}$ are the standard left-invariant one-form coordinate frame on the Lie group${\displaystyle SU(2)=S^3}$; that is, they obey${\displaystyle d{\sigma }_i=2{\sigma }_j\wedge {\sigma }_k}$ for${\displaystyle i,j,k=1,2,3}$ and cyclic permutations.

The corresponding local affine coordinates are${\displaystyle z_1=x+iy}$ and${\displaystyle z_2=z+it}$ then provide

with the usual abbreviations that${\displaystyle d{r}^2=dr\otimes dr}$ and${\displaystyle {\sigma }_k^2={\sigma }_k\otimes {\sigma }_k}$.

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

Thevierbeins 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:

${\displaystyle d{s}^2={\delta }_{ab}{e}^a\otimes e^b=e^0\otimes e^0+e^1\otimes e^1+e^2\otimes e^2+e^3\otimes e^3.}$

Given the vierbein, aspin connection can be computed; the Levi-Civita spin connection is the unique connection that istorsion-free and covariantly constant, namely, it is the one-form${\displaystyle {\omega }_b^a}$ that satisfies the torsion-free condition

${\displaystyle d{e}^a+{\omega }_b^a\wedge e^b=0}$

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

${\displaystyle {\omega }_{ab}=-{\omega }_{ba}}$

The above is readily solved; one obtains

Thecurvature 2-form is defined as

${\displaystyle R_b^a=d{\omega }_b^a+{\omega }_c^a\wedge {\omega }_b^c}$

and is constant:

TheRicci tensor in vierbein indexes is given by

${\displaystyle {\mathrm{Ric}}_c^a=R_{bcd}^a{\delta }^{bd}}$

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

${\displaystyle R_b^a=\frac{1}{2}{R}_{bcd}^a{e}^c\wedge e^d}$

The resultingRicci tensor is constant

${\displaystyle {\mathrm{Ric}}_{ab}=6{\delta }_{ab}}$

so that the resultingEinstein equation

${\displaystyle {\mathrm{Ric}}_{ab}-\frac{1}{2}{\delta }_{ab}R+\mathrm{\Lambda }{\delta }_{ab}=0}$

can be solved with thecosmological constant${\displaystyle \mathrm{\Lambda }=6}$.

TheWeyl tensor for Fubini–Study metrics in general is given by

${\displaystyle W_{abcd}=R_{abcd}-2\left({\delta }_{ac}{\delta }_{bd}-{\delta }_{ad}{\delta }_{bc}\right)}$

For then = 2 case, the two-forms

${\displaystyle W_{ab}=\frac{1}{2}{W}_{abcd}{e}^c\wedge e^d}$

are self-dual:

In then = 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 radiusR has sectional curvature${\displaystyle 1/R^2}$). However, forn > 1, the Fubini–Study metric does not have constant curvature. Its sectional curvature is instead given by the equation^[6]

${\displaystyle K(\sigma )=1+3⟨JX,Y{⟩}^2}$

where${\displaystyle \{X,Y\}\in T_p{\mathbf{C}\mathbf{P}}^n}$ is an orthonormal basis of the 2-plane σ, the mappingJ : TCPⁿ → TCPⁿ is thecomplex structure onCPⁿ, and${\displaystyle ⟨\cdot ,\cdot ⟩}$ is the Fubini–Study metric.

A consequence of this formula is that the sectional curvature satisfies${\displaystyle 1\le K(\sigma )\le 4}$ for all 2-planes${\displaystyle \sigma }$. The maximum sectional curvature (4) is attained at aholomorphic 2-plane — one for whichJ(σ) ⊂ σ — while the minimum sectional curvature (1) is attained at a 2-plane for whichJ(σ) is orthogonal to σ. For this reason, the Fubini–Study metric is often said to have "constantholomorphic sectional curvature" equal to 4.

This makesCPⁿ a (non-strict)quarter pinched manifold; a celebrated theorem shows that a strictly quarter-pinchedsimply connectedn-manifold must be homeomorphic to a sphere.

The Fubini–Study metric is also anEinstein metric in that it is proportional to its ownRicci tensor: there exists a constant${\displaystyle \mathrm{\Lambda }}$; such that for alli,j we have

${\displaystyle {\mathrm{Ric}}_{ij}=\mathrm{\Lambda }{g}_{ij}.}$

This implies, among other things, that the Fubini–Study metric remains unchanged up to a scalar multiple under theRicci flow. It also makesCPⁿ indispensable to the theory ofgeneral relativity, where it serves as a nontrivial solution to the vacuumEinstein field equations.

Thecosmological constant${\displaystyle \mathrm{\Lambda }}$ forCPⁿ is given in terms of the dimension of the space:

${\displaystyle {\mathrm{Ric}}_{ij}=2(n+1)g_{ij}.}$

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, if${\displaystyle |\psi ⟩}$ is aseparable state, so that it can be written as${\displaystyle |\psi ⟩=|{\psi }_A⟩\otimes |{\psi }_B⟩}$, then the metric is the sum of the metric on the subspaces:

${\displaystyle d{s}^2={d{s}_A}^2+{d{s}_B}^2}$

where${\displaystyle {d{s}_A}^2}$ and${\displaystyle {d{s}_B}^2}$ are the metrics, respectively, on the subspacesA andB.

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

${\displaystyle {\mathrm{\Gamma }}_{jk}^i=g^{i\overline{m}}\frac{\mathrm{\partial }{g}_{k\overline{m}}}{\mathrm{\partial }{z}^j}{\mathrm{\Gamma }}_{\overline{j}\overline{k}}^{\overline{i}}=g^{\overline{i}m}\frac{\mathrm{\partial }{g}_{\overline{k}m}}{\mathrm{\partial }{\overline{z}}^{\overline{j}}}}$

The Riemann tensor is also particularly simple:

${\displaystyle R_{i\overline{j}k\overline{l}}=g^{i\overline{m}}\frac{\mathrm{\partial }{\mathrm{\Gamma }}_{\overline{j}\overline{l}}^{\overline{m}}}{\mathrm{\partial }{z}^k}}$

TheRicci tensor is

${\displaystyle R_{\overline{i}j}=R_{\overline{i}\overline{k}j}^{\overline{k}}=-\frac{\mathrm{\partial }{\mathrm{\Gamma }}_{\overline{i}\overline{k}}^{\overline{k}}}{\mathrm{\partial }{z}^j}{R}_{i\overline{j}}=R_{ik\overline{j}}^k=-\frac{\mathrm{\partial }{\mathrm{\Gamma }}_{ik}^k}{\mathrm{\partial }{\overline{z}}^{\overline{j}}}}$

• Non-linear sigma model
• Kaluza–Klein theory
• Arakelov height

• Besse, Arthur L. (1987),Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York:Springer-Verlag, pp. xii+510,ISBN 978-3-540-15279-8
• Brody, D.C.; Hughston, L.P. (2001), "Geometric Quantum Mechanics",Journal of Geometry and Physics,38 (1):19–53,arXiv:quant-ph/9906086,Bibcode:2001JGP....38...19B,doi:10.1016/S0393-0440(00)00052-8,S2CID 17580350
• Griffiths, P.;Harris, J. (1994),Principles of Algebraic Geometry, Wiley Classics Library, Wiley Interscience, pp. 30–31,ISBN 0-471-05059-8
• Onishchik, A.L. (2001) [1994],"Fubini–Study metric",Encyclopedia of Mathematics,EMS Press.

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