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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Topological equivalence of gradient vectorfields
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by Douglas S. Shafer PDF
Trans. Amer. Math. Soc. 258 (1980), 113-126 Request permission

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.
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Additional 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