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 . 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 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 connected n-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 cosmological constant for CPn is given in terms of the dimension of the space: Product metric 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