Reeb spaces of smooth functions on manifolds II
The Reeb space of a continuous function on a topological space is the space of connected components of the level sets. In this paper we characterize those smooth functions on closed manifolds whose Reeb spaces have the structure of a finite graph. We also give several explicit examples of smooth fun...
Saved in:
| Published in: | Research in the mathematical sciences Vol. 11; no. 2 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cham
Springer International Publishing
01.06.2024
|
| Subjects: | |
| ISSN: | 2522-0144, 2197-9847 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | The Reeb space of a continuous function on a topological space is the space of connected components of the level sets. In this paper we characterize those smooth functions on closed manifolds whose Reeb spaces have the structure of a finite graph. We also give several explicit examples of smooth functions on closed manifolds such that they themselves or their Reeb spaces have some interesting properties. |
|---|---|
| AbstractList | The Reeb space of a continuous function on a topological space is the space of connected components of the level sets. In this paper we characterize those smooth functions on closed manifolds whose Reeb spaces have the structure of a finite graph. We also give several explicit examples of smooth functions on closed manifolds such that they themselves or their Reeb spaces have some interesting properties. |
| ArticleNumber | 24 |
| Author | Saeki, Osamu |
| Author_xml | – sequence: 1 givenname: Osamu orcidid: 0000-0003-1679-9948 surname: Saeki fullname: Saeki, Osamu email: saeki@imi.kyushu-u.ac.jp organization: Institute of Mathematics for Industry, Kyushu University |
| BookMark | eNp9j8tqwzAQRUVJoWmaH-jKP6Bm9LAlL0vowxAolOyFLEutgy0FyVk0X1-lyaqLrGYY5lzuuUczH7xF6JHAEwEQq8ShkgID5RiAswofb9CcklrgWnIxy3tJKQbC-R1aprQDACIqxhnM0erT2rZIe21sKoIr0hjC9F24gzdTH3y--WLUvndh6FLRNA_o1ukh2eVlLtD29WW7fsebj7dm_bzBhpViwq2jpXUgW8qtJVbwXMwwKyXjjkHX5a66Zka0VSVZV1OjqWhZy50wtCo5WyB5jjUxpBStU6af9KnRFHU_KALqpK7O6irHqT91dcwo_YfuYz_q-HMdYmco5Wf_ZaPahUP02fAa9Qsxd21v |
| CitedBy_id | crossref_primary_10_1016_j_topol_2025_109462 |
| Cites_doi | 10.1090/S0002-9947-1934-1501735-3 10.1007/s10587-015-0195-0 10.1007/b100393 10.1007/s10587-013-0034-0 10.1093/imrn/rnaa301 10.1112/jlms/2.Part_4.589 10.1112/jlms/s2-3.3.392 10.1515/ms-2016-0298 10.2298/FIL1907031G 10.1007/BFb0075066 10.13140/RG.2.2.27662.43844 10.1007/s00454-017-9957-9 10.1112/plms/s2-41.3.191 10.1093/qmath/haaa064 |
| ContentType | Journal Article |
| Copyright | The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| Copyright_xml | – notice: The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| DBID | AAYXX CITATION |
| DOI | 10.1007/s40687-024-00436-z |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 2197-9847 |
| ExternalDocumentID | 10_1007_s40687_024_00436_z |
| GrantInformation_xml | – fundername: Japan Society for the Promotion of Science grantid: JP22K18267, JP23H05437 funderid: http://dx.doi.org/10.13039/501100001691 |
| GroupedDBID | -EM 0R~ 406 5VS AAAVM AACDK AAHNG AAJBT AASML AATNV AATVU AAUYE AAYZH ABAKF ABDZT ABECU ABFTV ABJNI ABJOX ABKCH ABMQK ABSXP ABTEG ABTKH ABTMW ABXPI ACAOD ACDTI ACGFS ACHSB ACMLO ACOKC ACPIV ACZOJ ADHHG ADINQ ADKNI ADKPE ADTPH ADURQ ADYFF ADZKW AEFQL AEJRE AEMSY AESKC AFBBN AFQWF AGDGC AGJBK AGMZJ AGQEE AGRTI AHBYD AHSBF AHYZX AIAKS AIGIU AILAN AITGF AJZVZ ALMA_UNASSIGNED_HOLDINGS AMKLP AMXSW AMYLF AMYQR ASPBG AXYYD BGNMA C24 C6C CSCUP DPUIP EBLON EBS EJD FIGPU FINBP FNLPD FSGXE GGCAI GROUPED_DOAJ IKXTQ IWAJR J-C JZLTJ KOV LLZTM M4Y M~E NPVJJ NQJWS NU0 O9J OK1 PT4 ROL RSV SISQX SJYHP SNE SNPRN SOHCF SOJ SPISZ SRMVM SSLCW STPWE TSG UOJIU UTJUX UZXMN VFIZW Z88 ZMTXR AAYXX ABBRH ABDBE ABFSG ABRTQ ACSTC AEZWR AFDZB AFHIU AFOHR AHPBZ AHWEU AIXLP AMVHM ATHPR AYFIA CITATION |
| ID | FETCH-LOGICAL-c357t-bf25ef08b24ee1e74004c3e8834f30dd024a93c7b6683d92ca27b3b4f7c26543 |
| IEDL.DBID | RSV |
| ISICitedReferencesCount | 2 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=001183533300001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 2522-0144 |
| IngestDate | Tue Nov 18 22:28:57 EST 2025 Sat Nov 29 06:11:21 EST 2025 Fri Feb 21 02:41:33 EST 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 2 |
| Keywords | 54C30 57R45 Reeb space Reeb graph Smooth function Secondary 58K30 Primary 58K05 58K15 Peano continuum 57R70 |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c357t-bf25ef08b24ee1e74004c3e8834f30dd024a93c7b6683d92ca27b3b4f7c26543 |
| ORCID | 0000-0003-1679-9948 |
| ParticipantIDs | crossref_citationtrail_10_1007_s40687_024_00436_z crossref_primary_10_1007_s40687_024_00436_z springer_journals_10_1007_s40687_024_00436_z |
| PublicationCentury | 2000 |
| PublicationDate | 2024-06-01 |
| PublicationDateYYYYMMDD | 2024-06-01 |
| PublicationDate_xml | – month: 06 year: 2024 text: 2024-06-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | Cham |
| PublicationPlace_xml | – name: Cham |
| PublicationTitle | Research in the mathematical sciences |
| PublicationTitleAbbrev | Res Math Sci |
| PublicationYear | 2024 |
| Publisher | Springer International Publishing |
| Publisher_xml | – name: Springer International Publishing |
| References | SaekiOReeb spaces of smooth functions on manifoldsInt. Math. Res. Not.202220221187408768442584810.1093/imrn/rnaa301 SharkoVVAbout Kronrod–Reeb graph of a function on a manifoldMethods Funct. Anal. Topol.2006123893962279875 SaekiOTopology of Singular Fibers of Differentiable Maps2004BerlinSpringer GelbukhIThe co-rank of the fundamental group: the direct product, the first Betti number, and the topology of foliationsMath. Slovaca201767645656366074610.1515/ms-2016-0298 GelbukhILoops in Reeb graphs of n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-manifoldsDiscrete Comput. Geom.201859843863380230710.1007/s00454-017-9957-9 GelbukhIClose cohomologous Morse forms with compact leavesCzechoslovak Math. J.201363138515528307397510.1007/s10587-013-0034-0 WhitneyHAnalytic extensions of differentiable functions defined in closed setsTrans. Am. Math. Soc.1934366389150173510.1090/S0002-9947-1934-1501735-3 ReebGSur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numériqueC. R. Hebd. Seances Acad. Sci.1946222847849 GelbukhIApproximation of metric spaces by Reeb graphs: cycle rank of a Reeb graph, the co-rank of the fundamental group, and large components of level sets on Riemannian manifoldsFilomat20193320312049403635910.2298/FIL1907031G Bröcker, T., Jänich, K.: Introduction to Differential Topology. Cambridge University Press, Cambridge (Translated from the German by C.B. Thomas and M.J. Thomas) (1982) LevineHClassifying Immersions into R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R} }^{4}$$\end{document} Over Stable Maps of 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3$$\end{document}-Manifolds into R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R} }^2$$\end{document}1985BerlinSpringer WardAJThe topological characterisation of an open linear intervalProc. Lond. Math. Soc.1936S2–41191198157711010.1112/plms/s2-41.3.191 GelbukhICo-rank and Betti number of a groupCzechoslovak Math. J.201565140565567336044610.1007/s10587-015-0195-0 WillardSGeneral Topology1970ReadingAddison-Wesley Publishing Co. FranklinSPKrishnaraoGVOn the topological characterization of the real line: an addendumJ. Lond. Math. Soc.19712339228234110.1112/jlms/s2-3.3.392 KirwanFPeningtonGMorse theory without non-degeneracyQuart. J. Math.202172455514427139310.1093/qmath/haaa064 FranklinSPKrishnaraoGVOn the topological characterisation of the real lineJ. Lond. Math. Soc.19702258959126885610.1112/jlms/2.Part_4.589 GelbukhIOn the topology of the Reeb graphPublicationes Mathematicae Debrecen2024474469210.13140/RG.2.2.27662.43844 I Gelbukh (436_CR4) 2013; 63 O Saeki (436_CR13) 2004 SP Franklin (436_CR2) 1970; 2 SP Franklin (436_CR3) 1971; 2 I Gelbukh (436_CR5) 2015; 65 G Reeb (436_CR12) 1946; 222 436_CR1 I Gelbukh (436_CR7) 2018; 59 VV Sharko (436_CR15) 2006; 12 AJ Ward (436_CR16) 1936; S2–41 H Whitney (436_CR17) 1934; 36 O Saeki (436_CR14) 2022; 2022 H Levine (436_CR11) 1985 I Gelbukh (436_CR6) 2017; 67 F Kirwan (436_CR10) 2021; 72 I Gelbukh (436_CR8) 2019; 33 S Willard (436_CR18) 1970 I Gelbukh (436_CR9) 2024 |
| References_xml | – reference: GelbukhIThe co-rank of the fundamental group: the direct product, the first Betti number, and the topology of foliationsMath. Slovaca201767645656366074610.1515/ms-2016-0298 – reference: KirwanFPeningtonGMorse theory without non-degeneracyQuart. J. Math.202172455514427139310.1093/qmath/haaa064 – reference: WardAJThe topological characterisation of an open linear intervalProc. Lond. Math. Soc.1936S2–41191198157711010.1112/plms/s2-41.3.191 – reference: SharkoVVAbout Kronrod–Reeb graph of a function on a manifoldMethods Funct. Anal. Topol.2006123893962279875 – reference: GelbukhIClose cohomologous Morse forms with compact leavesCzechoslovak Math. J.201363138515528307397510.1007/s10587-013-0034-0 – reference: GelbukhILoops in Reeb graphs of n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-manifoldsDiscrete Comput. Geom.201859843863380230710.1007/s00454-017-9957-9 – reference: WhitneyHAnalytic extensions of differentiable functions defined in closed setsTrans. Am. Math. Soc.1934366389150173510.1090/S0002-9947-1934-1501735-3 – reference: FranklinSPKrishnaraoGVOn the topological characterization of the real line: an addendumJ. Lond. Math. Soc.19712339228234110.1112/jlms/s2-3.3.392 – reference: LevineHClassifying Immersions into R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R} }^{4}$$\end{document} Over Stable Maps of 3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3$$\end{document}-Manifolds into R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R} }^2$$\end{document}1985BerlinSpringer – reference: GelbukhIApproximation of metric spaces by Reeb graphs: cycle rank of a Reeb graph, the co-rank of the fundamental group, and large components of level sets on Riemannian manifoldsFilomat20193320312049403635910.2298/FIL1907031G – reference: WillardSGeneral Topology1970ReadingAddison-Wesley Publishing Co. – reference: ReebGSur les points singuliers d’une forme de Pfaff complètement intégrable ou d’une fonction numériqueC. R. Hebd. Seances Acad. Sci.1946222847849 – reference: SaekiOReeb spaces of smooth functions on manifoldsInt. Math. Res. Not.202220221187408768442584810.1093/imrn/rnaa301 – reference: GelbukhICo-rank and Betti number of a groupCzechoslovak Math. J.201565140565567336044610.1007/s10587-015-0195-0 – reference: Bröcker, T., Jänich, K.: Introduction to Differential Topology. Cambridge University Press, Cambridge (Translated from the German by C.B. Thomas and M.J. Thomas) (1982) – reference: GelbukhIOn the topology of the Reeb graphPublicationes Mathematicae Debrecen2024474469210.13140/RG.2.2.27662.43844 – reference: FranklinSPKrishnaraoGVOn the topological characterisation of the real lineJ. Lond. Math. Soc.19702258959126885610.1112/jlms/2.Part_4.589 – reference: SaekiOTopology of Singular Fibers of Differentiable Maps2004BerlinSpringer – volume: 36 start-page: 63 year: 1934 ident: 436_CR17 publication-title: Trans. Am. Math. Soc. doi: 10.1090/S0002-9947-1934-1501735-3 – volume: 65 start-page: 565 issue: 140 year: 2015 ident: 436_CR5 publication-title: Czechoslovak Math. J. doi: 10.1007/s10587-015-0195-0 – volume: 222 start-page: 847 year: 1946 ident: 436_CR12 publication-title: C. R. Hebd. Seances Acad. Sci. – volume-title: Topology of Singular Fibers of Differentiable Maps year: 2004 ident: 436_CR13 doi: 10.1007/b100393 – volume: 63 start-page: 515 issue: 138 year: 2013 ident: 436_CR4 publication-title: Czechoslovak Math. J. doi: 10.1007/s10587-013-0034-0 – volume: 2022 start-page: 8740 issue: 11 year: 2022 ident: 436_CR14 publication-title: Int. Math. Res. Not. doi: 10.1093/imrn/rnaa301 – volume-title: General Topology year: 1970 ident: 436_CR18 – ident: 436_CR1 – volume: 2 start-page: 589 issue: 2 year: 1970 ident: 436_CR2 publication-title: J. Lond. Math. Soc. doi: 10.1112/jlms/2.Part_4.589 – volume: 2 start-page: 392 issue: 3 year: 1971 ident: 436_CR3 publication-title: J. Lond. Math. Soc. doi: 10.1112/jlms/s2-3.3.392 – volume: 67 start-page: 645 year: 2017 ident: 436_CR6 publication-title: Math. Slovaca doi: 10.1515/ms-2016-0298 – volume: 33 start-page: 2031 year: 2019 ident: 436_CR8 publication-title: Filomat doi: 10.2298/FIL1907031G – volume: 12 start-page: 389 year: 2006 ident: 436_CR15 publication-title: Methods Funct. Anal. Topol. – volume-title: Classifying Immersions into $${\mathbb{R} }^{4}$$ Over Stable Maps of $$3$$-Manifolds into $${\mathbb{R} }^2$$ year: 1985 ident: 436_CR11 doi: 10.1007/BFb0075066 – year: 2024 ident: 436_CR9 publication-title: Publicationes Mathematicae Debrecen doi: 10.13140/RG.2.2.27662.43844 – volume: 59 start-page: 843 year: 2018 ident: 436_CR7 publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-017-9957-9 – volume: S2–41 start-page: 191 year: 1936 ident: 436_CR16 publication-title: Proc. Lond. Math. Soc. doi: 10.1112/plms/s2-41.3.191 – volume: 72 start-page: 455 year: 2021 ident: 436_CR10 publication-title: Quart. J. Math. doi: 10.1093/qmath/haaa064 |
| SSID | ssj0001763430 ssib053846527 |
| Score | 2.3157678 |
| Snippet | The Reeb space of a continuous function on a topological space is the space of connected components of the level sets. In this paper we characterize those... |
| SourceID | crossref springer |
| SourceType | Enrichment Source Index Database Publisher |
| SubjectTerms | Applications of Mathematics Computational Mathematics and Numerical Analysis Mathematics Mathematics and Statistics |
| Title | Reeb spaces of smooth functions on manifolds II |
| URI | https://link.springer.com/article/10.1007/s40687-024-00436-z |
| Volume | 11 |
| WOSCitedRecordID | wos001183533300001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVAVX databaseName: Springer Journals New Starts & Take-Overs Collection customDbUrl: eissn: 2197-9847 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0001763430 issn: 2522-0144 databaseCode: RSV dateStart: 20141201 isFulltext: true titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22 providerName: Springer Nature |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LS8NAEF60etCDb7G-2IM3XUz3kU2OIhZ7sEgt0lvY7AMETaSpHvrrnd0mgYIU9D7ZLLMz880-5huErrhQBrJWQXydIuFUJUQBDBMuRa5EZDxmhWYTcjhMJpP0uS4Kq5rX7s2VZIjUbbEbQA84BGAKCbzpZL6ONgDupH_IN3p5bawIPJjHoua2DCct4EI8NB2hkGwQv4Woq2d-H3YZoZavRwPq9Hf_N989tFNnmfhuYRb7aM0WB2j7qaVorQ7R7cjaHEM8gUCBS4erjxIWDXucC6aIywJ7bgxXvpsKDwZHaNx_GN8_krp7AtFMyBnJHRXWRUlOubU9K72zamaThHHHImNgYiplWuZxnDCTUq2ozFnOndTUF5weo05RFvYE4RjSCmOtSFxPcWUiz-gOaaVOjWM6prSLeo3CMl0zi_sGF-9Zy4kcdJHBL7Ogi2zeRdftN58LXo2V0jeNjrPax6oV4qd_Ez9DWzQskz9bOUed2fTLXqBN_T17q6aXwbh-AFnzxJo |
| linkProvider | Springer Nature |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LSwMxEB60CurBt1ifOXjT4DaPfRxFLC22RbRIb0s2DxDqrnSrh_56k3S3IEhB79ndMJmZ70uy8w3AFeNCWdbKsatTxIyIGAsLw5hFPBM8UA6zfLOJaDCIR6PkqSoKK-u_3esrSZ-pF8VuFnpsQFhMwV43Hc9WYY0Ry_DdHv3ltfYiG8Es5JW2pT9psSHEfNMRYskGdluIqnrm99f-RKif16Meddo7_5vvLmxXLBPdzd1iD1Z0vg9b_YVEa3kAt89aZ8jmE5soUGFQ-V7YRUMO57wroiJHThvDFGNVom73EIbth-F9B1fdE7CkPJrizBCuTRBnhGnd0pELVkl1HFNmaKCUnZhIqIyyMIypSogUJMpoxkwkiSs4PYJGXuT6GFBoaYXSmsemJZhQgVN0t7RSJspQGRLShFZtsFRWyuKuwcU4XWgie1uk9pOpt0U6a8L14pmPua7G0tE3tY3TKsbKJcNP_jb8EjY6w34v7XUHj6ewSfySuXOWM2hMJ5_6HNbl1_StnFx4R_sGm0XHfg |
| linkToPdf | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1JSwMxFH5oFdGDu1jXHLxp6DTLLEdRi0UtRYt4G2aygFBnSmf00F9vks4UC1IQ75lJeHkv38vyfQ_ggvFEmqyVY8tTxIwkIU4MDGMW8DThnrSY5YpNBL1e-PYW9X-w-N1r9_pKcsppsCpNWdkaSd2aEd8MDJngMPiCnYY6nizDCrPP5ex-_eW19igTzcznlc6lO3Ux4cRcARJiEg9stxMVk-b3386j1fxVqUOgztb_x74Nm1X2ia6n7rIDSyrbhY2nmXRrsQetZ6VSZNYZs4CgXKPiIzeTiSz-ORdFeYasZobOh7JA3e4-DDp3g5t7XFVVwILyoMSpJlxpL0wJU6qtAhvEgqowpExTT0ozsCSiIkh9P6QyIiIhQUpTpgNBLBH1ABpZnqlDQL5JN6RSPNTthCXSs0rvJt0UkdRU-IQ0oV0bLxaV4rgtfDGMZ1rJzhax6TJ2tognTbicfTOa6m0sbH1V2zuuYq9Y0Pzob83PYa1_24kfu72HY1gnbsbs8csJNMrxpzqFVfFVvhfjM-dz35vp0Fk |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Reeb+spaces+of+smooth+functions+on+manifolds+II&rft.jtitle=Research+in+the+mathematical+sciences&rft.au=Saeki%2C+Osamu&rft.date=2024-06-01&rft.issn=2522-0144&rft.eissn=2197-9847&rft.volume=11&rft.issue=2&rft_id=info:doi/10.1007%2Fs40687-024-00436-z&rft.externalDBID=n%2Fa&rft.externalDocID=10_1007_s40687_024_00436_z |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2522-0144&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2522-0144&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2522-0144&client=summon |