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Mat. Sb., 1999, Volume 190, Number 3, Pages 29–88 (Mi msb392)  

This article is cited in 15 scientific papers (total in 15 papers)

Realization of smooth functions on surfaces as height functions

E. A. Kudryavtseva

M. V. Lomonosov Moscow State University

Abstract: A criterion describing all functions with finitely many critical points on two-dimensional surfaces that can be height functions corresponding to some immersions of the surface in three-dimensional Euclidean space is established. It is proved that each smooth deformation of a Morse function on the surface can be realized as the deformation of the height function induced by a suitable deformation of the immersion of the surface in $\mathbb R^3$. A new proof of the well-known result on the path connectedness of the space of all smooth immersions of a two-dimensional sphere in $\mathbb R^3$ obtained. A new description of an eversion of a two-dimensional sphere in $\mathbb R^3$ is given. Generalizations of S. Matveev's result on the connectedness of the space of Morse functions with fixed numbers of minima and maxima on a closed surface are established.

DOI: https://doi.org/10.4213/sm392

Full text: PDF file (727 kB)
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English version:
Sbornik: Mathematics, 1999, 190:3, 349–405

Bibliographic databases:

UDC: 515.162.6+515.164.63+515.148+515.164.174
MSC: Primary 57R42, 57R52; Secondary 58F07
Received: 26.02.1998

Citation: E. A. Kudryavtseva, “Realization of smooth functions on surfaces as height functions”, Mat. Sb., 190:3 (1999), 29–88; Sb. Math., 190:3 (1999), 349–405

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Maksymenko, S, “Path-components of Morse mappings spaces of surfaces”, Commentarii Mathematici Helvetici, 80:3 (2005), 655  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    2. Maksymenko, S, “Homotopy types of stabilizers and orbits of morse functions on surfaces”, Annals of Global Analysis and Geometry, 29:3 (2006), 241  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    3. Nicolaescu, LI, “Counting Morse functions on the 2-sphere”, Compositio Mathematica, 144:5 (2008), 1081  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    4. Kudryavtseva E.A., “Uniform Morse lemma and isotopy criterion for Morse functions on surfaces”, Moscow Univ. Math. Bull., 64:4 (2009), 150–158  crossref  mathscinet  zmath  elib  scopus
    5. E. A. Kudryavtseva, D. A. Permyakov, “Framed Morse functions on surfaces”, Sb. Math., 201:4 (2010), 501–567  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. Masumoto Ya., Saeki O., “A Smooth Function on a Manifold with Given Reeb Graph”, Kyushu J Math, 65:1 (2011), 75–84  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    7. Maksymenko S.I., “Deformations of Circle-Valued Morse Functions on Surfaces”, Ukrainian Math J, 62:10 (2011), 1577–1584  crossref  mathscinet  isi  scopus  scopus  scopus
    8. Morishita F., Saeki O., “Height functions on surfaces with three critical values”, J Math Soc Japan, 63:1 (2011), 153–162  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    9. E. A. Kudryavtseva, “The Topology of Spaces of Morse Functions on Surfaces”, Math. Notes, 92:2 (2012), 219–236  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    10. E. A. Kudryavtseva, “Special framed Morse functions on surfaces”, Moscow University Mathematics Bulletin, 67:4 (2012), 151–157  mathnet  crossref  mathscinet
    11. E. A. Kudryavtseva, “Connected components of spaces of Morse functions with fixed critical points”, Moscow University Mathematics Bulletin, 67:1 (2012), 1–10  mathnet  crossref  mathscinet
    12. E. A. Kudryavtseva, “On the homotopy type of spaces of Morse functions on surfaces”, Sb. Math., 204:1 (2013), 75–113  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. Maksymenko S.I., Feshchenko B.G., “Homotopic Properties of the Spaces of Smooth Functions on a 2-Torus”, Ukr. Math. J., 66:9 (2015), 1346–1353  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    14. Kudryavtseva E.A., “Topology of the spaces of functions with prescribed singularities on surfaces”, Dokl. Math., 93:3 (2016), 264–266  crossref  mathscinet  zmath  isi  scopus
    15. Feshchenko B., “Actions of Finite Groups and Smooth Functions on Surfaces”, Methods Funct. Anal. Topol., 22:3 (2016), 210–219  mathscinet  zmath  isi
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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