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Transactions of the American Mathematical Society

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

 

 

Topological equivalence of gradient vectorfields


Author: Douglas S. Shafer
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
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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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DOI: https://doi.org/10.1090/S0002-9947-1980-0554322-6
Keywords: Gradient vectorfield, topological equivalence
Article copyright: © Copyright 1980 American Mathematical Society