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The tensor $T_f$ vanishes if and only if the map $f$ is weakly conformal. The norm $|T_f|$ is a quantity which is a measure of conformality of $f$ at each point. We are concerned with maps which are critical points of the functional $\\Phi (f)$ $=$ ${\\displaystyle \\int_M |T_f|^2dv_g}$. We call such maps {\\it C-stationary map"},"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":"1209.4249","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-09-19T14:14:09Z","cross_cats_sorted":[],"title_canon_sha256":"fbb73e6b86f5ce4909a8f923e429c6400ae572f767fcc61199b2cfe5a24bdee0","abstract_canon_sha256":"b554a6f2eb00a99ed8da2cee2657322134070da3c70a758ae4f251e0ea6dabd8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:46:04.790531Z","signature_b64":"EH8flHaRF/oXnjD4FN1b34wxHD8g9d152bKUBR10oezUMYGvB+VjueSWdKzSrUdVB08rwnZFRZAp1/9a8qEwDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9f874dd3c3493ee4045d0bed98304405429203b0698d52b53a07645275478f5","last_reissued_at":"2026-05-18T02:46:04.789937Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:46:04.789937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak conformality of stable stationary maps for a functional related to conformality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Nobumitsu Nakauchi, Shigeo Kawai","submitted_at":"2012-09-19T14:14:09Z","abstract_excerpt":"Let $(M, g)$, $(N, h)$ be compact Riemannian manifolds without boundary, and let $f$ be a smooth map from $M$ into $N$. We consider a covariant symmetric tensor $T_f$ $=$ ${\\displaystyle f^*h - \\frac{1}{m} |df|^2 g}$, where $f^*h$ denotes the pull-back metric of $h$ by $f$. The tensor $T_f$ vanishes if and only if the map $f$ is weakly conformal. The norm $|T_f|$ is a quantity which is a measure of conformality of $f$ at each point. We are concerned with maps which are critical points of the functional $\\Phi (f)$ $=$ ${\\displaystyle \\int_M |T_f|^2dv_g}$. 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