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 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 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 [Z0:...:Zn], there is a unique set of n coordinates (z1,...,zn) such that provided Z0 ≠ 0; specifically, zj = Zj/Z0. The (z1,...,zn) form an affine coordinate system for CPn in the coordinate patch U0 = {Z0 ≠ 0}. One can develop an affine coordinate system in any of the coordinate patches Ui = {Zi ≠ 0} by dividing instead by Zi in the obvious manner. The n+1 coordinate patches Ui cover CPn, and it is possible to give the metric explicitly in terms of the affine coordinates (z1,...,zn) on Ui. 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. The metric can be derived from the following Kähler