Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Topological equivalence of gradient vectorfields
HTML articles powered by AMS MathViewer

by Douglas S. Shafer
Trans. Amer. Math. Soc. 258 (1980), 113-126
DOI: https://doi.org/10.1090/S0002-9947-1980-0554322-6

Abstract:

This paper is a study of the behavior of the topological equivalence class of the planar gradient vectorfield $X = {\operatorname {grad} _g} V$, in a neighborhood of a degenerate singularity of V, as g varies over all Riemannian metrics. It is shown that under simple restrictions on V the topological equivalence class of X is determined by its first nonvanishing jet, and that only finitely many equivalence classes occur (for fixed V). In this case, when the degree of the first nonvanishing jet of V is less than five, necessary and sufficient conditions for change in equivalence class are given, both in terms of the coefficients of the homogeneous part of V and geometrically in terms of its level curves. A catalogue of possible phase portraits, up to topological equivalence, is included. Necessary conditions are given for change in higher degree.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 58F14
  • Retrieve articles in all journals with MSC: 58F14
Bibliographic Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 258 (1980), 113-126
  • MSC: Primary 58F14
  • DOI: https://doi.org/10.1090/S0002-9947-1980-0554322-6
  • MathSciNet review: 554322