Proceedings of the Japan Academy, Series A, Mathematical Sciences

A study of curvature using infinitesimals

Satoshi Koike, Tzee-Char Kuo, and Laurentiu Paunescu

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Motivated by a profound observation of A’Campo we investigate the behaviour of the curvature of $f(z,w)=c, f\in \mathbf{C}\{z,w\}, |c|$ small, along infinitesimals. We use the language of infinitesimals as introduced in [2]. Along the way we introduce the important notion of gradient canyon, and prove several theorems in which this notion plays the key role. In this paper we give three such theorems and mention several other facts, to be published elsewhere.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 88, Number 5 (2012), 70-74.

First available in Project Euclid: 7 May 2012

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Zentralblatt MATH identifier

Primary: 14HXX 32SXX 58K60: Deformation of singularities
Secondary: 58K40: Classification; finite determinacy of map germs

Newton-Piuseux field curvature Lojasiewicz exponent gradient canyon Newton-Puiseux infinitesimals


Koike, Satoshi; Kuo, Tzee-Char; Paunescu, Laurentiu. A study of curvature using infinitesimals. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 5, 70--74. doi:10.3792/pjaa.88.70.

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