Curvature-Driven Shear Deformation: A Unified Geometric Framework for Gravitational Memory and Geophysical Flows

Curvature-Driven Shear Deformation: A Unified Geometric Framework for Gravitational Memory and Geophysical Flows | 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 Curvature-Driven Shear Deformation: A Unified Geometric Framework for Gravitational Memory and Geophysical Flows Asheer Ahmed This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7188590/v2 This work is licensed under a CC BY 4.0 License Status: Posted Version 2 posted You are reading this latest preprint version Show more versions Abstract This work addresses a fundamental question: How does intrinsic curvature alter the definition and evolution of shear deformation in curved geometries? Classical approaches based on flat-space assumptions fail due to coordinate dependence, path-dependent parallelism (holonomy), and topological constraints, creating significant gaps in understanding systems ranging from relativistic spacetimes to planetary atmospheres. We resolve this by developing a unified geometric framework that intrinsically defines shear deforma tion on Riemannian and Lorentzian manifolds. Using coordinate-invariant methods—including the Lie derivative of the metric, covariant derivatives, and Jacobi fields—we construct a curvature-compatible strain tensor and derive its evolution equation directly coupled to the Riemann curvature tensor. Our approach yields three transformative outcomes: First, in General Relativity, we unify tidal forces, geodesic deviation, and gravitational wave memory through explicit spacetime strain-curvature coupling. Second, for elastic membranes on spheres, we quantify geometric frustration caused by topological obstruc tion, explaining anomalous stress relaxation. Third, in geophysical flows, we predict curvature-induced instabilities like jet stream localization using the Einstein tensor, revealing phenomena absent in flat-space models. By synthesizing differential geometry, continuum mechanics, and relativity, this work establishes fun damental tools for deformation analysis in curved spaces, with direct applications to gravitational wave detection, atmospheric science, and soft matter engineering. Shear deformation Riemannian geometry strain tensor gravitational wave memory ge ometric frustration non-Euclidean flows curvature constraints holonomy continuum mechanics Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 2 posted You are reading this latest preprint version Show more versions 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-7188590","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":489524324,"identity":"27c1ab02-b04e-4ffe-89d8-d14f89267ae0","order_by":0,"name":"Asheer Ahmed","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABCUlEQVRIiWNgGAWjYFACHjCZAMQGh/8w2ABpxsYDxGoxPMDDkAbS0kC0FmOglsNgHl4t/P1nDz74uMcuT7e9ecMBiZrzdmvbDwNtqbGJxqVF4kZesuGMZ8nFZmeOFRwwOHY7eduZRKCWY2m5Dbj03OAxk+Y5wJy47UaOwYEEttvJZgeAWhgbDuPUIn/+jPnvPwfqE7fdf2Nw4MC/c8lm5x/i12JwIMeMmeHAYaAtPAYHG9sO2JndIGCL4Y0cY8meA8cTt51JKzjM2JecYHYDaEsCHr/InT9j+OHHgerEbccPb/7M8M3O3ux8+sMHH2pscHsfHSSCVSYQqxwE7ElRPApGwSgYBSMDAABblnCzrTrCCgAAAABJRU5ErkJggg==","orcid":"","institution":"University of Delhi","correspondingAuthor":true,"prefix":"","firstName":"Asheer","middleName":"","lastName":"Ahmed","suffix":""}],"badges":[],"createdAt":"2025-07-22 15:23:16","currentVersionCode":2,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7188590/v2","doiUrl":"https://doi.org/10.21203/rs.3.rs-7188590/v2","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":90192619,"identity":"5818e670-45ac-4648-888b-eecc388f718b","added_by":"auto","created_at":"2025-08-29 16:08:37","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3228251,"visible":true,"origin":"","legend":"","description":"","filename":"CURVATUREDRIVENSHEARDEFORMATION.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7188590/v2_covered_1d16be81-c983-48e7-ad2f-01999f400c2a.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eCurvature-Driven Shear Deformation: A Unified Geometric Framework for Gravitational Memory and Geophysical Flows\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"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":"Shear deformation, Riemannian geometry, strain tensor, gravitational wave memory, ge ometric frustration, non-Euclidean flows, curvature constraints, holonomy, continuum mechanics","lastPublishedDoi":"10.21203/rs.3.rs-7188590/v2","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7188590/v2","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis work addresses a fundamental question: How does intrinsic curvature alter the definition and evolution \u0026nbsp;of shear deformation in curved geometries? Classical approaches based on flat-space assumptions fail due \u0026nbsp;to coordinate dependence, path-dependent parallelism (holonomy), and topological constraints, creating \u0026nbsp;significant gaps in understanding systems ranging from relativistic spacetimes to planetary atmospheres. \u0026nbsp;We resolve this by developing a unified geometric framework that intrinsically defines shear deforma tion on Riemannian and Lorentzian manifolds. Using coordinate-invariant methods—including the Lie \u0026nbsp;derivative of the metric, covariant derivatives, and Jacobi fields—we construct a curvature-compatible \u0026nbsp;strain tensor and derive its evolution equation directly coupled to the Riemann curvature tensor. \u0026nbsp;Our approach yields three transformative outcomes: First, in General Relativity, we unify tidal forces, \u0026nbsp;geodesic deviation, and gravitational wave memory through explicit spacetime strain-curvature coupling. \u0026nbsp;Second, for elastic membranes on spheres, we quantify geometric frustration caused by topological obstruc tion, explaining anomalous stress relaxation. Third, in geophysical flows, we predict curvature-induced \u0026nbsp;instabilities like jet stream localization using the Einstein tensor, revealing phenomena absent in flat-space \u0026nbsp;models. \u0026nbsp;By synthesizing differential geometry, continuum mechanics, and relativity, this work establishes fun damental tools for deformation analysis in curved spaces, with direct applications to gravitational wave \u0026nbsp;detection, atmospheric science, and soft matter engineering.\u003c/p\u003e","manuscriptTitle":"Curvature-Driven Shear Deformation: A Unified Geometric Framework for Gravitational Memory and Geophysical Flows","msid":"","msnumber":"","nonDraftVersions":[{"code":2,"date":"2025-08-29 15:52:27","doi":"10.21203/rs.3.rs-7188590/v2","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}},{"code":1,"date":"2025-07-23 06:00:50","doi":"10.21203/rs.3.rs-7188590/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"2a93afe2-a114-412a-be8a-bf768fb36340","owner":[],"postedDate":"August 29th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-08-07T15:23:42+00:00","versionOfRecord":[],"versionCreatedAt":"2025-08-29 15:52:27","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v2","identity":"rs-7188590","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7188590","identity":"rs-7188590","version":["v2"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}