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Let $\mathcal{H}_{n,k}(\Sigma)$ be the space of degree $n\geq 1$ holomorphicmaps from a compact Riemann surface $\Sigma $ to $\mathbb{C}P^k$. In the case$\Sigma=S^2$ and $n=1$, the $L^2$ metric on $\mathcal{H}_{1,k}(S^2)$ wascomputed exactly by Speight. In this paper, the Ricci curvature tensor and thescalar curvature on $\mathcal{H}_{1,k}(S^2)$ are determined explicitly for$k\geq 2$. An exact direct computation of the Einstein-Hilbert action withrespect to the $L^2$ metric on $\mathcal{H}_{1,k}(S^2)$ is made and shown tocoincide with a formula conjectured by Baptista.

The Einstein-Hilbert action of the space of holomorphic maps from S^2 to CP^k