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

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

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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

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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• http://mi.mathnet.ru/eng/msb392
• https://doi.org/10.4213/sm392
• http://mi.mathnet.ru/eng/msb/v190/i3/p29

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