Skip to main content
SpringerLink
Log in

Functions with Nondegenerate Critical Points on the Boundary of the Surface

  • Published:
Ukrainian Mathematical Journal Aims and scope

We prove an analog of the Morse theorem in the case where the critical point belongs to the boundary of an n-dimensional manifold and find the least number of critical points for the Morse functions defined on the surfaces whose critical points coincide with the critical points of their restriction to the boundary.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Milnor and A. H. Wallace, Differential Topology [Russian translation], Mir, Moscow (1972).

  2. A. Jankowski and R. Rubinsztein, “Functions with nondegenerate critical points on manifolds with boundary,” Comment. Math. Prace Mat., 16, 99–112 (1972).

    MathSciNet  MATH  Google Scholar 

  3. G. Reeb, “Sur les points singulies de une forme de Pfaff completement integrable ou de une function numerique,” Comp. Rend. Hebdomadaires Seaces Acad. Sci., 222, 847–849 (1954).

    MathSciNet  MATH  Google Scholar 

  4. A. S. Kronrod, “On functions of two variables,” Usp. Mat. Nauk, 5, No. 1(35), 24–112 (1950).

  5. V. V. Sharko, “Functions on surfaces,” in: Some Problems of Contemporary Mathematics [in Russian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 25 (1998), pp. 408–434.

  6. A.V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, and Classification [in Russian], Vol. 1, Izd. Dom “Udmurskii Universitet,” Izhevsk (1999).

  7. S. I. Maksymenko, “Equivalence of m-functions on surfaces,” Ukr. Mat. Zh., 51, No. 8, 1129–1135 (1999); English translation : Ukr. Math. J., 51, No. 8, 1275–1281 (1999).

  8. O. O. Pryshlyak, “Functions of general position on surfaces with boundary,” Visn. Kyiv. Univ., Mat. Mekh., No. 20, 77–79 (2008).

  9. Y. Matsumoto, An Introduction to Morse Theory, American Mathematical Society, Providence, RI (1985).

    Google Scholar 

  10. M. Borodzik, A. Nemethi, and A. Ranicki, Morse Theory for Manifolds with Boundary, Preprint arXiv:1207. 3066v4[math. GT] (2014).

  11. V. I. Arnold, “The calculus of snakes and the combinatorics of Bernoulli, Euller and Springer numbers of Coxeter groups,” Usp. Mat. Nauk, 47, No. 1(283), 3–45 (1992).

Download references

Author information

Authors and Affiliations

  1. Shevchenko Kyiv National University, Kyiv, Ukraine

    B. I. Hladysh & O. O. Pryshlyak

Authors
  1. B. I. Hladysh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. O. Pryshlyak
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 1, pp. 28–37, January, 2016.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hladysh, B.I., Pryshlyak, O.O. Functions with Nondegenerate Critical Points on the Boundary of the Surface. Ukr Math J 68, 29–40 (2016). https://doi.org/10.1007/s11253-016-1206-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-016-1206-5

Search

Navigation