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(4π) −d T d k=1 K t 2k ,t 2k+1 (x)dt involves products of d sectional curvatures K ij (x) averaged over the space T ∼ O(2d) of all" /> <meta property="article:author" content="https://harvard.academia.edu/OliverKnill" /> <meta name="description" content="We look at the functional γ(M) = M K(x)dV (x) for compact Riemannian 2dmanifolds M , where K(x) = (2d)!(d!) −1 (4π) −d T d k=1 K t 2k ,t 2k+1 (x)dt involves products of d sectional curvatures K ij (x) averaged over the space T ∼ O(2d) of all" /> <title>(PDF) A Dehn type quantity for Riemannian manifolds</title> <link rel="canonical" href="https://www.academia.edu/125874994/A_Dehn_type_quantity_for_Riemannian_manifolds" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 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{"work":{"id":125874994,"created_at":"2024-11-26T23:03:34.238-08:00","from_world_paper_id":261374488,"updated_at":"2024-12-09T15:12:21.472-08:00","_data":{"publisher":"Cornell University","grobid_abstract":"We look at the functional γ(M) = M K(x)dV (x) for compact Riemannian 2dmanifolds M , where K(x) = (2d)!(d!) −1 (4π) −d T d k=1 K t 2k ,t 2k+1 (x)dt involves products of d sectional curvatures K ij (x) averaged over the space T ∼ O(2d) of all orthonormal frames t = (t 1 ,. .. , t 2d). A discrete version γ d (M) with K d (x) = (d!) −1 (4π) −d σ d k=1 K σ(2k−1),σ(2k) sums over all permutations σ of {1,. .. , 2d}. Unlike Euler characteristic which by Gauss-Bonnet-Chern is M K GBC dV = χ(M), the quantities γ or γ d are in general metric dependent. We are interested in δ(M) = γ(M) − χ(M) because if M has curvature sign e, then γ(M)e d and γ d (M) are positive while χ(M)e d \u003e 0 is only conjectured. We compute γ d in a few concrete examples like 2d-spheres, the 4-manifold CP 2 , the 6 manifold SO(4) or the 8-manifold SU (3).","publication_date":"2020,5,25","publication_name":"arXiv (Cornell University)","grobid_abstract_attachment_id":"119841153"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"low","language":"en","title":"A Dehn type quantity for Riemannian manifolds","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [24636766]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":119841153,"attachmentType":"pdf"}"><img alt="First page of “A Dehn type quantity for Riemannian manifolds”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/119841153/mini_magick20241127-1-71hr8.png?1732691031" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">A Dehn type quantity for Riemannian manifolds</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="24636766" href="https://harvard.academia.edu/OliverKnill"><img alt="Profile image of Oliver Knill" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/24636766/6662359/7528121/s65_oliver.knill.jpg" />Oliver Knill</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2020, arXiv (Cornell University)</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">30 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 125874994; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=125874994"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">We look at the functional γ(M) = M K(x)dV (x) for compact Riemannian 2dmanifolds M , where K(x) = (2d)!(d!) −1 (4π) −d T d k=1 K t 2k ,t 2k+1 (x)dt involves products of d sectional curvatures K ij (x) averaged over the space T ∼ O(2d) of all orthonormal frames t = (t 1 ,. .. , t 2d). A discrete version γ d (M) with K d (x) = (d!) −1 (4π) −d σ d k=1 K σ(2k−1),σ(2k) sums over all permutations σ of {1,. .. , 2d}. Unlike Euler characteristic which by Gauss-Bonnet-Chern is M K GBC dV = χ(M), the quantities γ or γ d are in general metric dependent. We are interested in δ(M) = γ(M) − χ(M) because if M has curvature sign e, then γ(M)e d and γ d (M) are positive while χ(M)e d > 0 is only conjectured. We compute γ d in a few concrete examples like 2d-spheres, the 4-manifold CP 2 , the 6 manifold SO(4) or the 8-manifold SU (3).</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":119841153,"attachmentType":"pdf","workUrl":"https://www.academia.edu/125874994/A_Dehn_type_quantity_for_Riemannian_manifolds"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":119841153,"attachmentType":"pdf","workUrl":"https://www.academia.edu/125874994/A_Dehn_type_quantity_for_Riemannian_manifolds"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas" data-impression-entity-id="125874994" data-impression-entity-type="2" data-impression-source="signup-banner"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. 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A discrete version γ d (M) with K d (x) = (d!) −1 (4π) −d σ d k=1 K σ(2k−1),σ(2k) sums over all permutations σ of {1,. .. , 2d}. Unlike Euler characteristic which by Gauss-Bonnet-Chern is M K GBC dV = χ(M), the quantities γ or γ d are in general metric dependent. We are interested in δ(M) = γ(M) − χ(M) because if M has curvature sign e, then γ(M)e d and γ d (M) are positive while χ(M)e d > 0 is only conjectured. We compute γ d in a few concrete examples like 2d-spheres, the 4-manifold CP 2 , the 6 manifold SO(4) or the 8-manifold SU (3).</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Dehn type invariant for Riemannian manifolds","attachmentId":119841150,"attachmentType":"pdf","work_url":"https://www.academia.edu/125874993/A_Dehn_type_invariant_for_Riemannian_manifolds","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/125874993/A_Dehn_type_invariant_for_Riemannian_manifolds"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="99022551" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/99022551/A_curvature_identity_on_a_6_dimensional_Riemannian_manifold_and_its_applications">A curvature identity on a~6-dimensional Riemannian manifold and its applications</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="250427083" href="https://independent.academia.edu/EUHYunhee">Yunhee EUH</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Czechoslovak Mathematical Journal, 2017</p><p class="ds-related-work--abstract ds2-5-body-sm">We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. Moreover, some applications of the curvature identity are given. We also define a generalization of harmonic manifolds to study the Lichnerowicz conjecture for a harmonic manifold "a harmonic manifold is locally symmetric" and provide another proof of the Lichnerowicz conjecture refined by Ledger for the 4-dimensional case under a slightly more general setting.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A curvature identity on a~6-dimensional Riemannian manifold and its applications","attachmentId":100218979,"attachmentType":"pdf","work_url":"https://www.academia.edu/99022551/A_curvature_identity_on_a_6_dimensional_Riemannian_manifold_and_its_applications","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/99022551/A_curvature_identity_on_a_6_dimensional_Riemannian_manifold_and_its_applications"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="99157849" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/99157849/Curvature_measures_of_pseudo_Riemannian_manifolds">Curvature measures of pseudo-Riemannian manifolds</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="244282753" href="https://independent.academia.edu/DFaifman">Dmitry Faifman</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal für die reine und angewandte Mathematik (Crelles Journal)</p><p class="ds-related-work--abstract ds2-5-body-sm">The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric ( 0 , 2 ) {(0,2)} -tensors. More precisely, we construct a family of generalized curvature measures attached to such manifolds, extending the Riemannian Lipschitz–Killing curvature measures introduced by Federer. We then show that they behave naturally under isometric immersions, in particular they do not depend on the ambient signature. Consequently, we extend Theorema Egregium to surfaces equipped with a generic metric of changing signature, and more generally, establish the existence as distributions of intrinsically defined Lipschitz–Killing curvatures for such manifolds of arbitrary dimension. This includes in particular the scalar curvature and the Chern–Gauss–Bonnet integrand. Finally, we deduce a Chern–Gauss–Bonnet theorem for pseudo-Riemannian manifolds with generic boundary.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Curvature measures of pseudo-Riemannian manifolds","attachmentId":100319166,"attachmentType":"pdf","work_url":"https://www.academia.edu/99157849/Curvature_measures_of_pseudo_Riemannian_manifolds","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/99157849/Curvature_measures_of_pseudo_Riemannian_manifolds"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="124606353" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/124606353/On_index_expectation_curvature_for_manifolds">On index expectation curvature for manifolds</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="99076177" href="https://harvard.academia.edu/OKnill">Oliver Knill</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2020</p><p class="ds-related-work--abstract ds2-5-body-sm">Index expectation curvature K(x) = E[i f (x)] on a compact Riemannian 2dmanifold M is an expectation of Poincaré-Hopf indices i f (x) and so satisfies the Gauss-Bonnet relation M K(x) dV (x) = χ(M). Unlike the Gauss-Bonnet-Chern integrand, these curvatures are in general non-local. We show that for small 2d-manifolds M with boundary embedded in a parallelizable 2d-manifold N of definite sectional curvature sign e, an index expectation K(x) with definite sign e d exists. The function K(x) is constructed as a product k K k (x) of sectional index expectation curvature averages E[i k (x)] of a probability space of Morse functions f for which i f (x) = i k (x), where the i k are independent and so uncorrelated.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On index expectation curvature for manifolds","attachmentId":118801384,"attachmentType":"pdf","work_url":"https://www.academia.edu/124606353/On_index_expectation_curvature_for_manifolds","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/124606353/On_index_expectation_curvature_for_manifolds"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="99022546" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/99022546/A_Curvature_Identity_on_a_4_Dimensional_Riemannian_Manifold">A Curvature Identity on a 4-Dimensional Riemannian Manifold</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="250427083" href="https://independent.academia.edu/EUHYunhee">Yunhee EUH</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Results in Mathematics, 2011</p><p class="ds-related-work--abstract ds2-5-body-sm">We give a curvature identity derived from the generalized Gauss-Bonnet formula for 4-dimensional compact oriented Riemannian manifolds. We prove that the curvature identity holds on any 4-dimensional Riemannian manifold which is not necessarily compact. We also provide some applications of the identity.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Curvature Identity on a 4-Dimensional Riemannian Manifold","attachmentId":100219003,"attachmentType":"pdf","work_url":"https://www.academia.edu/99022546/A_Curvature_Identity_on_a_4_Dimensional_Riemannian_Manifold","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/99022546/A_Curvature_Identity_on_a_4_Dimensional_Riemannian_Manifold"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="99022557" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/99022557/4_dimensional_Riemannian_manifolds">4-dimensional Riemannian manifolds</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="250427083" href="https://independent.academia.edu/EUHYunhee">Yunhee EUH</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2016</p><p class="ds-related-work--abstract ds2-5-body-sm">We shall prove the universality of the curvature identity for the 4-dimensional Riemannian manifold using a different method than that used by Gilkey, Park, and Sekigawa [5].</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"4-dimensional Riemannian manifolds","attachmentId":100219006,"attachmentType":"pdf","work_url":"https://www.academia.edu/99022557/4_dimensional_Riemannian_manifolds","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/99022557/4_dimensional_Riemannian_manifolds"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="99022550" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/99022550/A_Remark_Concerning_Universal_Curvature_Identities_on_4_DIMENSIONAL_Riemannian_Manifolds">A Remark Concerning Universal Curvature Identities on 4-DIMENSIONAL Riemannian Manifolds</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="250427083" href="https://independent.academia.edu/EUHYunhee">Yunhee EUH</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Bulletin of the Korean Mathematical Society, 2012</p><p class="ds-related-work--abstract ds2-5-body-sm">We shall prove the universality of the curvature identity for the 4-dimensional Riemannian manifold using a different method than that used by Gilkey, Park, and Sekigawa [5].</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Remark Concerning Universal Curvature Identities on 4-DIMENSIONAL Riemannian Manifolds","attachmentId":100219008,"attachmentType":"pdf","work_url":"https://www.academia.edu/99022550/A_Remark_Concerning_Universal_Curvature_Identities_on_4_DIMENSIONAL_Riemannian_Manifolds","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/99022550/A_Remark_Concerning_Universal_Curvature_Identities_on_4_DIMENSIONAL_Riemannian_Manifolds"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="71027616" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/71027616/Compact_Manifolds_with_Positive_%CE%932_CURVATURE">Compact Manifolds with Positive Γ2-CURVATURE</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="57021813" href="https://independent.academia.edu/BorisBotvinnik">Boris Botvinnik</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2016</p><p class="ds-related-work--abstract ds2-5-body-sm">Abstract. The Schouten tensor A of a Riemannian manifold (M, g) provides important scalar curvature invariants σk, that are the symmetric functions on the eigenvalues of A, where, in particular, σ1 coincides with the standard scalar curvature Scal(g). Our goal here is to study compact manifolds with positive Γ2-curvature, i.e., when σ1(g)&gt; 0 and σ2(g)&gt; 0. In particular, we prove that a 3-connected non-string manifold M admits a positive Γ2-curvature metric if and only if it admits a positive scalar curvature metric. Also we show that any finitely presented group pi can always be realised as the fundamental group of a closed manifold of positive Γ2-curvature and of arbitrary dimension greater than or equal to six.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Compact Manifolds with Positive Γ2-CURVATURE","attachmentId":80541417,"attachmentType":"pdf","work_url":"https://www.academia.edu/71027616/Compact_Manifolds_with_Positive_%CE%932_CURVATURE","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/71027616/Compact_Manifolds_with_Positive_%CE%932_CURVATURE"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="95333202" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/95333202/A_New_3_dimensional_Curvature_Integral_Formula_for_PL_manifolds_of_Non_positive_Curvature">A New 3-dimensional Curvature Integral Formula for PL-manifolds of Non-positive Curvature</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="254080504" href="https://independent.academia.edu/JosEscobar2">Jos Escobar</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Communications in Analysis and Geometry, 2003</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we derive a new curvature integral formula for 3-dimensional piecewise linear manifolds with singularities. Among other things, we also present a sharp isoperimetric inequality for 3-dimensional PL-manifolds of non-positive curvature by using this new curvature integral formula. Let Ω be a smooth compact domain in a smooth Riemannian manifold, and GK ∂Ω represent the Gauss-Kronecker curvature (i.e., the determinant of the second fundamental form) of the boundary of Ω, ∂Ω. A well-known Theorem of Chern-Lashof [CL] states that for any compact convex smooth domain Ω in R n , the total Gauss-Kronecker curvature of its boundary satisfies ∂Ω GK ∂Ω dA = vol n−1 (S n−1) where S n−1 is the unit (n − 1)-dimensional sphere in the n-dimensional Euclidean space. It has been conjectured by various authors that for any compact convex smooth domain Ω in a Cartan-Hadamard manifold M n , the total Gauss-Kronecker curvature of its boundary satisfies ∂Ω GK ∂Ω dA ≥ vol n−1 (S n−1). (0.1) In fact, for a compact surface Σ in a 3-dimensional smooth Cartan-Hadamard manifold M 3 , the classical Gauss Theorem states K Σ − K M 3 | Σ = GK Σ , (0.2) where K Σ (resp. K M 3) is the sectional curvature of Σ (resp. M 3). It follows from the equation (0.2) and the Gauss-Bonnet formula that if ∂Ω is</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A New 3-dimensional Curvature Integral Formula for PL-manifolds of Non-positive Curvature","attachmentId":97543954,"attachmentType":"pdf","work_url":"https://www.academia.edu/95333202/A_New_3_dimensional_Curvature_Integral_Formula_for_PL_manifolds_of_Non_positive_Curvature","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/95333202/A_New_3_dimensional_Curvature_Integral_Formula_for_PL_manifolds_of_Non_positive_Curvature"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="99022556" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/99022556/A_Curvature_Identity_on_a_4_Dimensional_Riemannian">A Curvature Identity on a 4-Dimensional Riemannian</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="250427083" href="https://independent.academia.edu/EUHYunhee">Yunhee EUH</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2016</p><p class="ds-related-work--abstract ds2-5-body-sm">We give a curvature identity derived from the generalized Gauss-Bonnet formula for 4-dimensional compact oriented Riemannian manifolds. We prove that the curvature identity holds on any 4-dimensional Riemannian manifold which is not necessarily compact. We also provide some applications of the identity.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Curvature Identity on a 4-Dimensional Riemannian","attachmentId":100219007,"attachmentType":"pdf","work_url":"https://www.academia.edu/99022556/A_Curvature_Identity_on_a_4_Dimensional_Riemannian","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/99022556/A_Curvature_Identity_on_a_4_Dimensional_Riemannian"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":119841153,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":119841153,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_119841153" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. 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Weyl curvature invariant</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="286300721" href="https://independent.academia.edu/MLLabbi">M. L. Labbi</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2004</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Manifolds with positive second H. 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href="https://www.academia.edu/81043663/Related_aspects_of_positivity_in_Riemannian_geometry">Related aspects of positivity in Riemannian geometry</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="22784614" href="https://sbsuny.academia.edu/DennisSullivan">Dennis Sullivan</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Differential Geometry, 1987</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Related aspects of positivity in Riemannian geometry","attachmentId":87224703,"attachmentType":"pdf","work_url":"https://www.academia.edu/81043663/Related_aspects_of_positivity_in_Riemannian_geometry","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span 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href="https://uniroma1.academia.edu/AndreaSambusetti">Andrea Sambusetti</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematische Annalen, 1998</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"An obstruction to the existence of Einstein metrics on 4-manifolds","attachmentId":50678289,"attachmentType":"pdf","work_url":"https://www.academia.edu/2304389/An_obstruction_to_the_existence_of_Einstein_metrics_on_4_manifolds","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/2304389/An_obstruction_to_the_existence_of_Einstein_metrics_on_4_manifolds"><span 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