PROOF OF THE HOPF CONJECTURE

PROOF OF THE HOPF CONJECTURE | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article PROOF OF THE HOPF CONJECTURE Athoumane Niang, Thierno Seck, Adama Thiandoum This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3888921/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The aim of this note is to show that a metric induced by $\mathbb{R}^{6+k}$ on the graph of k-smooth functions on $S2 × S2$ does not have a positive sectional curvature. According to the remark of O. Perdomo in [11], this result solves the Hopf Conjecture. Mathematics Subject Classification 2010: 53A10, 53C42, 53C50. Metrics induced by graph of functions mooth functions Sectional curvature Hopf conjecture. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3888921","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":268928518,"identity":"2b93cec6-09ef-4ebe-86d8-6468326f54cf","order_by":0,"name":"Athoumane Niang","email":"data:image/png;base64,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","orcid":"","institution":"Cheikh Anta Diop University","correspondingAuthor":true,"prefix":"","firstName":"Athoumane","middleName":"","lastName":"Niang","suffix":""},{"id":268928519,"identity":"8f04c780-6bc4-4f03-a61f-799f5257c4ff","order_by":1,"name":"Thierno Seck","email":"","orcid":"","institution":"Cheikh Anta Diop University","correspondingAuthor":false,"prefix":"","firstName":"Thierno","middleName":"","lastName":"Seck","suffix":""},{"id":268928520,"identity":"83d7e4ad-2d71-4578-9e01-ce44682ea306","order_by":2,"name":"Adama Thiandoum","email":"","orcid":"","institution":"Cheikh Anta Diop University","correspondingAuthor":false,"prefix":"","firstName":"Adama","middleName":"","lastName":"Thiandoum","suffix":""}],"badges":[],"createdAt":"2024-01-22 20:14:10","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3888921/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3888921/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":53497109,"identity":"4971931b-7e77-45ee-b6bb-236053623eb6","added_by":"auto","created_at":"2024-03-26 17:14:03","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":194396,"visible":true,"origin":"","legend":"","description":"","filename":"Athoumanenewversion23012024.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3888921/v1_covered_27e828d0-d65e-4718-a530-250d6886c8ab.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"PROOF OF THE HOPF CONJECTURE","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Metrics induced by graph of functions, mooth functions, Sectional curvature, Hopf conjecture.","lastPublishedDoi":"10.21203/rs.3.rs-3888921/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3888921/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe aim of this note is to show that a metric induced by $\\mathbb{R}^{6+k}$ on the graph of k-smooth functions on $S2 × S2$ does not have a positive sectional curvature. 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