Advertisement

Ukrainian Mathematical Journal

, Volume 49, Issue 10, pp 1548–1558 | Cite as

Vector fields with a given set of singular points

  • A. O. Prishlyak
Article

Abstract

Theorems on the existence of vector fields with given sets of indexes of isolated singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a two-dimensional manifold, an index of an isolated singular point of the gradient field is not greater than one.

Keywords

Vector Field Singular Point Gradient Field Smooth Manifold Euler Characteristic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Poincaré, On Curves Determined by Differential Equations [Russian translation], OGIZ, Moscow-Leningrad (1947).Google Scholar
  2. 2.
    H. Hopf, “Vektorfelder in n-dimensionalen Mannigfaltigkeiten,” Math. Ann., 96, 209–221 (1926).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, Charlottesville (1965).zbMATHGoogle Scholar
  4. 4.
    M. W. Hirsch, Differential Topology, Springer-Verlag, New York-Heidelberg-Berlin (1976).zbMATHGoogle Scholar
  5. 5.
    V. I. Arnol'd Ordinary Differential Equations [in Russian], Nauka, Moscow (1971).Google Scholar
  6. 6.
    S. Smale, “On gradient dynamical systems,” Ann. Math. 74, No. 1, 199–206 (1961).CrossRefMathSciNetGoogle Scholar

Copyright information

© Plenum Publishing Corporation 1998

Authors and Affiliations

  • A. O. Prishlyak

There are no affiliations available

Personalised recommendations