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In this paper, the Ricci curvature tensor and the scalar curvature on $\\mathcal{H}_{1,k}(S^2)$ are determined explicitly for $k\\geq 2$. An exact direct computation of the Einstein-Hilbert action with respect to the $L^2$ metric on $\\mathcal{H}_{1,k}(S^2)$ is made and shown to coincide with a formula conjectured by Baptista."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1303.2036","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-03-08T15:58:08Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"d157327e806599b18ee7cdcb8ecc44c8c41341fa2bffda1b7da69ae49e5aa52e","abstract_canon_sha256":"4d6a1b1bc8ac9920a01d5e85cbb5d75c2ab046bba3673f2c5ab3ecd8053ebdb0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:51:09.601165Z","signature_b64":"PsSNSi5XS5DeRYFW1JHUlaQFhF0DV240X0yLU08R7fjLGwO+AzqeRRGmTX6YzWDozYh8AY79zLKchXYTJcR4AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4841227564435fb35b4540b783110d823942d9040b24a079caf6f4350a6eca25","last_reissued_at":"2026-05-18T01:51:09.600520Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:51:09.600520Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Einstein-Hilbert action of the space of holomorphic maps from S^2 to CP^k","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"L.S. 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