A
Hermitian form in (the vector space) Cn+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, CPn 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
(2n+1)-sphere. In
algebraic geometry, one uses a normalization making CPn a
Hodge manifold.
Specifically, one may define CPn to be the space consisting of all complex lines in Cn+1, i.e., the quotient of Cn+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 Cn+1\{0} as a complex
line bundle over the base space CPn. (In fact this is the so-called
tautological bundle over CPn.) A point of CPn is thus identified with an equivalence class of (n+1)-tuples [Z0,...,Zn] modulo nonzero complex rescaling; the Zi are called
homogeneous coordinates of the point.
Furthermore, one may realize this quotient mapping in two steps: since multiplication by a nonzero complex scalar z = Reiθ can 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 mapping Cn+1 → CPn splits into two pieces.
where step (a) is a quotient by the dilation Z ~ RZ for R ∈ R+, the multiplicative group of
positive real numbers, and step (b) is a quotient by the rotations Z ~ eiθZ.
The result of the quotient in (a) is the real hypersphere S2n+1 defined by the equation |Z|2 = |Z0|2 + ... + |Zn|2 = 1. The quotient in (b) realizes CPn = S2n+1/S1, where S1 represents the group of rotations. This quotient is realized explicitly by the famous
Hopf fibrationS1 → S2n+1 → CPn, the fibers of which are among the
great circles of .
As a metric quotient
When a quotient is taken of a
Riemannian manifold (or
metric space in general), care must be taken to ensure that the quotient space is endowed with a
metric that is well-defined. For instance, if a group G acts on a Riemannian manifold (X,g), then in order for the
orbit spaceX/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 fields we must have g(Xh,Yh) = g(X,Y).
The standard
Hermitian metric on Cn+1 is given in the standard basis by
whose realification is the standard
Euclidean metric on R2n+2. This metric is not invariant under the diagonal action of C*, so we are unable to directly push it down to CPn in the quotient. However, this metric is invariant under the diagonal action of S1 = 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 CPn = S2n+1/S1, where carries 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 CPn with homogeneous coordinates , there is a unique set of n coordinates such that
provided ; specifically, . The form an
affine coordinate system for CPn in the coordinate patch . One can develop an affine coordinate system in any of the coordinate patches by dividing instead by in the obvious manner. The n+1 coordinate patches cover CPn, and it is possible to give the metric explicitly in terms of the affine coordinates on . The coordinate derivatives define a frame of the holomorphic tangent bundle of CPn, in terms of which the Fubini–Study metric has Hermitian components
where |z|2 = |z1|2 + ... + |zn|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 action will 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.
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 ds2 apparently defines a tensor on the total space of the tautological bundle Cn+1\{0}. It is to be understood properly as a tensor on CPn by pulling it back along a holomorphic section σ of the tautological bundle of CPn. 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.
where the are the
Dolbeault operators.
The pullback of this is clearly independent of the choice of holomorphic section. The quantity log|Z|2 is the
Kähler potential (sometimes called the Kähler scalar) of CPn.
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
Here, is the
complex conjugate of . The appearance of in the denominator is a reminder that and likewise 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 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, CP1 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
When n = 1, there is a diffeomorphism given by
stereographic projection. This leads to the "special" Hopf fibration S1 → S3 → S2. When the Fubini–Study metric is written in coordinates on CP1, its restriction to the real tangent bundle yields an expression of the ordinary "round metric" of radius 1/2 (and
Gaussian curvature 4) on S2.
Namely, if z = x + iy is the standard affine coordinate chart on the
Riemann sphereCP1 and x = r cos θ, y = r sin θ are polar coordinates on C, then a routine computation shows
where is the round metric on the unit 2-sphere. Here φ, θ are "mathematician's
spherical coordinates" on S2 coming from the stereographic projection r tan(φ/2) = 1, tan θ = y/x. (Many physics references interchange the roles of φ and θ.)
The Fubini–Study metric on the
complex projective planeCP2 has been proposed as a
gravitational instanton, the gravitational analog of an
instanton.[5][3] The metric, the connection form and the curvature are readily computed, once suitable real 4D coordinates are established. Writing for real Cartesian coordinates, one then defines polar coordinate one-forms on the
4-sphere (the
quaternionic projective line) as
The are the standard left-invariant one-form coordinate frame on the Lie group ; that is, they obey for cyclic.
The corresponding local affine coordinates are and then provide
with the usual abbreviations that and .
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, a
spin connection can be computed; the Levi-Civita spin connection is the unique connection that is
torsion-free and covariantly constant, namely, it is the one-form that satisfies the torsion-free condition
and is covariantly constant, which, for spin connections, means that it is antisymmetric in the vierbein indexes:
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]
where is an orthonormal basis of the 2-plane σ, J : TCPn → TCPn is the
complex structure on CPn, and is the Fubini–Study metric.
A consequence of this formula is that the sectional curvature satisfies for all 2-planes . The maximum sectional curvature (4) is attained at a
holomorphic 2-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 CPn a (non-strict)
quarter pinched manifold; a celebrated theorem shows that a strictly quarter-pinched
simply connectedn-manifold must be homeomorphic to a sphere.
The Fubini–Study metric is also an
Einstein metric in that it is proportional to its own
Ricci 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 CPn indispensable to the theory of
general relativity, where it serves as a nontrivial solution to the vacuum
Einstein field equations.
The common notions of separability apply for the Fubini–Study metric. More precisely, the metric is separable on the natural product of projective spaces, the
Segre embedding. That is, if is a
separable state, so that it can be written as , then the metric is the sum of the metric on the subspaces:
where and are the metrics, respectively, on the subspaces A and B.
Connection and curvature
The fact that the metric can be derived from the Kähler potential means that the
Christoffel symbols and the curvature tensors contain a lot of symmetries, and can be given a particularly simple form:[7] The Christoffel symbols, in the local affine coordinates, are given by
^G. Fubini, "Sulle metriche definite da una forma Hermitiana", (1904) Atti del Reale Istituto Veneto di Scienze, Lettere ed Arti, 63 pp. 501–513
^Study, E. (1905). "Kürzeste Wege im komplexen Gebiet". Mathematische Annalen (in German). 60 (3). Springer Science and Business Media LLC: 321–378.
doi:
10.1007/bf01457616.
ISSN0025-5831.
S2CID120961275.
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,
ISBN978-3-540-15279-8