Abstract
Let Hn,k(Σ) be the space of degree n≥1 holomorphic maps from a compact Riemann surface Σ to CPk. In the case Σ=S2 and n=1, the L2 metric on H1,k(S2) was computed exactly by Speight. In this paper, the Ricci curvature tensor and the scalar curvature on H1,k(S2) are determined explicitly for k≥2. An exact direct computation of the Einstein–Hilbert action with respect to the L2 metric on H1,k(S2) is made and shown to coincide with a formula conjectured by Baptista.