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 sphere CP1 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 Kähler form is Choosing as vierbeins and , the Kähler form simplifies to Applying the Hodge star to the Kähler form, one obtains implying that K is harmonic. The n = 2 case The Fubini–Study metric on the complex projective plane CP2 has been proposed as a gravitational instanton, the gravitational analog of an instanton. 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