FubiniStudy

The Fubini-Study metric is a canonical Riemannian metric on complex projective space CP^n, invariant under the natural action of U(n+1). It is the unique homogeneous Kähler metric up to scale with constant holomorphic sectional curvature 4, making CP^n a compact Hermitian symmetric space.

Construction and description: The metric is induced from the standard Hermitian inner product on C^{n+1} via the Hopf fibration S^{2n+1} → CP^n. In homogeneous coordinates [z0:...:zn], the Kähler potential is K = log ∑ |z_i|^2, and the associated Kähler form is ω_FS = i ∂∂̄ K. In inhomogeneous coordinates w_j = z_j/z_0 (j = 1,...,n), the metric components are g_{i\bar{j}} = ∂^2/∂w_i ∂\bar{w}_j log(1 + ∑ |w_k|^2).

Distance: For unit vectors u,v ∈ C^{n+1}, the distance between the corresponding points [u],[v] in CP^n is d_FS([u],[v]) = arccos( |⟨u, v⟩| ), where ⟨.,.⟩ is the standard Hermitian inner product.

Properties: The metric on CP^n with FS is compact, simply connected, and complete. The isometry group is essentially PU(n+1). The metric has constant holomorphic sectional curvature 4, and the Ricci curvature satisfies Ric = (n+1) g under the common normalization.

Applications: In quantum mechanics, the Fubini-Study metric describes the geometry of pure states, with the distance reflecting transition probabilities. In complex differential geometry and algebraic geometry, it serves as a fundamental example of a Kähler metric on a projective space and informs studies of projective embeddings and geometric quantization.