{"id":"https://openalex.org/W7225464415","doi":"https://doi.org/10.5281/zenodo.24026009","title":"Curvature Generates Geometry: Unified Formulas and Systematic Derivations for Spherical, Euclidean, and Hyperbolic Geometries","display_name":"Curvature Generates Geometry: Unified Formulas and Systematic Derivations for Spherical, Euclidean, and Hyperbolic Geometries","publication_year":2026,"publication_date":"2026-11-07","ids":{"openalex":"https://openalex.org/W7225464415","doi":"https://doi.org/10.5281/zenodo.24026009"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.24026009","doi":"https://doi.org/10.5281/zenodo.24026009","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24026009","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.24026009","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5138061132","display_name":"Ni Chuangao","orcid":"https://orcid.org/0009-0000-1987-8401","observed_orcids":["https://orcid.org/0009-0000-1987-8401"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"NI, Chuangao","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0009-0000-1987-8401","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5138061132"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.9667999744415283,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.9667999744415283,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.021700000390410423,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.008500000461935997,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/comparison-geometry","display_name":"comparison geometry","score":0.7739999890327454},{"id":"https://openalex.org/keywords/spherical-geometry","display_name":"spherical geometry","score":0.7739999890327454},{"id":"https://openalex.org/keywords/hyperbolic-geometry","display_name":"hyperbolic geometry","score":0.8809999823570251},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.8709999918937683},{"id":"https://openalex.org/keywords/second-order-linear-differential-equations","display_name":"second-order linear differential equations","score":0.6759999990463257},{"id":"https://openalex.org/keywords/jacobi-fields","display_name":"Jacobi fields","score":0.9269999861717224},{"id":"https://openalex.org/keywords/exponential-map","display_name":"exponential map","score":0.8349999785423279},{"id":"https://openalex.org/keywords/conjugate-points","display_name":"conjugate points","score":0.7070000171661377},{"id":"https://openalex.org/keywords/distance-metrics","display_name":"distance measures","score":0.4009999930858612},{"id":"https://openalex.org/keywords/volume","display_name":"volume","score":0.3529999852180481},{"id":"https://openalex.org/keywords/steiner-formula","display_name":"Steiner formula","score":0.7319999933242798},{"id":"https://openalex.org/keywords/tubular-neighborhoods","display_name":"tubular neighborhoods","score":0.7429999709129333},{"id":"https://openalex.org/keywords/spherical-harmonics","display_name":"spherical harmonics","score":0.8519999980926514},{"id":"https://openalex.org/keywords/circle-geometry","display_name":"circle geometry","score":0.7990000247955322},{"id":"https://openalex.org/keywords/isoperimetric-inequality","display_name":"isoperimetric inequality","score":0.9950000047683716},{"id":"https://openalex.org/keywords/rigidity-theorems","display_name":"rigidity theorems","score":0.9539999961853027}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8285999894142151},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.5174000263214111},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4869000017642975},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.47859999537467957},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.44830000400543213},{"id":"https://openalex.org/C62884695","wikidata":"https://www.wikidata.org/wiki/Q184871","display_name":"Euler's formula","level":2,"score":0.4081000089645386},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.4027999937534332},{"id":"https://openalex.org/C2776716506","wikidata":"https://www.wikidata.org/wiki/Q5161158","display_name":"Conjugate points","level":2,"score":0.35100001096725464},{"id":"https://openalex.org/C2779668893","wikidata":"https://www.wikidata.org/wiki/Q2250422","display_name":"Constant curvature","level":3,"score":0.3407000005245209},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.33640000224113464},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3343999981880188},{"id":"https://openalex.org/C29001434","wikidata":"https://www.wikidata.org/wiki/Q8084","display_name":"Trigonometry","level":2,"score":0.3165999948978424},{"id":"https://openalex.org/C99564226","wikidata":"https://www.wikidata.org/wiki/Q505628","display_name":"Hyperboloid","level":2,"score":0.31450000405311584},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.30970001220703125},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.3093999922275543},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.30410000681877136},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.28529998660087585},{"id":"https://openalex.org/C108598597","wikidata":"https://www.wikidata.org/wiki/Q124255","display_name":"Conic section","level":2,"score":0.2689000070095062},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.2619999945163727},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.26010000705718994},{"id":"https://openalex.org/C178009071","wikidata":"https://www.wikidata.org/wiki/Q93344","display_name":"Trigonometric functions","level":2,"score":0.25679999589920044},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.25429999828338623},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.25189998745918274}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.24026009","doi":"https://doi.org/10.5281/zenodo.24026009","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24026009","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.24026010","doi":"https://doi.org/10.5281/zenodo.24026010","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24026010","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.5281/zenodo.24026009","doi":"https://doi.org/10.5281/zenodo.24026009","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.24026009","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7225251919","https://openalex.org/W1841239832","https://openalex.org/W7055469396","https://openalex.org/W7223143335","https://openalex.org/W7077152020","https://openalex.org/W2792086643","https://openalex.org/W7222407149","https://openalex.org/W4233159126","https://openalex.org/W4254068076","https://openalex.org/W2491355897"],"abstract_inverted_index":{"This":[0],"paper":[1,94,265,338,365,403],"establishes":[2],"a":[3,151,249,252,256,287,321,407,436],"complete":[4,152,340,437],"unified":[5,49,104,161,295,424],"formula":[6,106,237,297],"system":[7],"from":[8,155,440],"the":[9,25,35,48,54,58,66,77,80,85,96,103,110,120,135,156,167,176,184,210,217,268,272,294,304,313,328,331,352,355,364,384,389,399,420,426,433,441],"second-order":[10],"linear":[11],"ordinary":[12],"differential":[13],"equation":[14,369],"$y''+ky=0$":[15],"to":[16,158,302,443],"constant-curvature":[17,59,281,393,411],"geometry,":[18,20],"comparison":[19,145,159,219,233,247,284,357,414,444],"and":[21,82,90,102,116,123,129,164,175,181,187,193,198,204,209,215,226,234,241,255,283,290,316,319,335,358,388,413,432],"rigidity":[22,228,359,378,395],"theorems,":[23],"with":[24,345],"framework":[26],"of":[27,84,126,137,170,221,263,280,323,354,383,392,423,429,435],"**curvature-generated":[28],"geometry**":[29],"(i.e.,":[30],"curvature":[31,45,177],"generates":[32,47,57,69],"geometry)":[33],"as":[34,43,109,119,271,298,406],"organizing":[36],"thread.":[37],"The":[38,93,147,260,337,402],"core":[39,261,278,350],"generating":[40,165,269],"chain":[41,154,270,439],"is":[42,372,396],"follows:":[44],"$k$":[46],"trigonometric":[50,162],"functions":[51,163],"$\\sin_k,\\cos_k$":[52],"through":[53],"ODE;":[55],"$\\sin_k$":[56],"metric":[60,67],"$g_k=dr^2+\\sin_k^2(r)\\,d\\theta^2$":[61],"in":[62,134,232,374,419],"geodesic":[63],"polar":[64],"coordinates;":[65],"then":[68],"all":[70,277],"downstream":[71],"geometric":[72],"quantities,":[73],"including":[74],"Jacobi":[75,179,311],"fields,":[76,312],"exponential":[78,185,314],"map,":[79,315],"Hessian":[81],"Laplacian":[83],"distance":[86,190],"function,":[87],"volume,":[88,195],"trigonometry,":[89,201],"circle":[91,207],"geometry.":[92],"takes":[95],"complex":[97,205],"parameter":[98],"$\\alpha$":[99],"(with":[100],"$k=-\\alpha^2$)":[101],"Euler":[105,296],"$e^{\\alpha":[107],"r}=\\cos_k(r)+\\alpha\\sin_k(r)$":[108],"algebraic":[111,300],"anchor,":[112],"unifying":[113],"spherical,":[114],"Euclidean,":[115],"hyperbolic":[117],"geometries":[118,333],"imaginary,":[121],"zero,":[122],"real":[124],"branches":[125],"$\\alpha$,":[127],"respectively,":[128],"provides":[130,339],"chapter-by-chapter":[131],"systematic":[132,334,408,421],"derivations":[133],"form":[136,139,150],"\u201cunified":[138],"+":[140,144],"three-geometry":[141,246,325,430],"explicit":[142],"forms":[143],"tables.\u201d":[146],"13":[148],"chapters":[149],"derivation":[153,428,438],"ODE":[157,442],"rigidity:":[160],"geometry;":[166,415],"full":[168],"set":[169],"elementary":[171],"functions;":[172],"Christoffel":[173],"symbols":[174],"tensor;":[178],"fields":[180],"conjugate":[182],"points;":[183],"map":[186],"pullback":[188],"metric;":[189],"Hessian,":[191,223],"Laplacian,":[192,224],"convexity;":[194],"tubular":[196],"neighborhoods,":[197],"Steiner":[199],"formulas;":[200],"spherical":[202],"harmonics,":[203],"trigonometry;":[206],"geometry":[208,282,285,412],"isoperimetric":[211],"inequality;":[212],"small-curvature":[213],"expansions":[214],"$k$-responses;":[216],"five":[218],"theorems":[220,229,360],"Rauch,":[222],"Bishop\u2013Gromov,":[225],"Toponogov;":[227],"at":[230],"equality":[231],"quantitative":[235],"stability;":[236],"tables,":[238,248],"dependency":[239],"trees,":[240],"open":[242],"problems.":[243],"Appendices":[244],"provide":[245],"complex-parameter":[250],"manual,":[251],"symbol":[253],"index,":[254],"Euclidean-limit":[257],"operation":[258],"manual.":[259],"contributions":[262],"this":[264],"are:":[266],"**taking":[267],"principal":[273],"thread,":[274],"systematically":[275],"linking":[276],"formulas":[279],"into":[286],"directed,":[288],"derivable,":[289],"self-contained":[291,397],"system**;":[292],"**using":[293],"an":[299],"anchor":[301],"unify":[303],"Pythagorean":[305],"identity,":[306],"addition":[307],"formulas,":[308,425,431],"power":[309],"series,":[310],"model":[317],"angles**;":[318],"**adopting":[320],"methodology":[322],"item-by-item":[324,427],"realization,":[326],"making":[327],"correspondences":[329],"among":[330],"three":[332],"visible**.":[336],"proofs,":[341],"or":[342],"proof":[343],"ideas":[344],"key":[346,356],"steps,":[347],"for":[348,370,410],"each":[349],"formula;":[351],"proofs":[353],"are":[361],"closed":[362],"within":[363],"(the":[366],"shape-operator":[367],"Riccati":[368],"Hessian/Laplacian":[371],"inlined":[373],"Chapter":[375,385],"11,":[376],"Bishop\u2013Gromov":[377],"relies":[379],"on":[380],"Step":[381],"5":[382],"11":[386],"comparison,":[387],"two-dimensional":[390],"case":[391],"global":[394],"via":[398],"covering-map":[400],"route).":[401],"can":[404],"serve":[405],"reference":[409],"its":[416],"value":[417],"lies":[418],"organization":[422],"establishment":[434],"rigidity.":[445]},"counts_by_year":[],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2026-11-08T00:00:00"}
