This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.
[1][2] 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.
In algebraic geometry, one uses a normalization making CPn a Hodge manifold.
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.
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 = R eiθ 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
This quotient is realized explicitly by the famous Hopf fibration S1 → S2n+1 → CPn, the fibers of which are among the great circles of
For instance, if a group G acts on a Riemannian manifold (X,g), then in order for the orbit 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 fields
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.
cover CPn, and it is possible to give the metric explicitly in terms of the affine coordinates
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.
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 An expression is also possible in the notation of homogeneous coordinates, commonly used to describe projective varieties of algebraic geometry: Z = [Z0:...:Zn].
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 ds2 apparently defines a tensor on the total space of the tautological bundle Cn+1\{0}.
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.
is the standard notation for a point in the projective space CPn in homogeneous coordinates.
in the space, the distance (length of a geodesic) between them is or, equivalently, in projective variety notation, Here,
Namely, if z = x + iy is the standard affine coordinate chart on the Riemann sphere CP1 and x = r cos θ, y = r sin θ are polar coordinates on C, then a routine computation shows where
[5][3] The metric, the connection form and the curvature are readily computed, once suitable real 4D coordinates are established.
are the standard left-invariant one-form coordinate frame on the Lie group
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 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 the cosmological constant
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 connected n-manifold must be homeomorphic to a sphere.
; 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.
for CPn is given in terms of the dimension of the space: 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.
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 The Riemann tensor is also particularly simple: The Ricci tensor is