Journal of the Mathematical Society of Japan

A conjecture in relation to Loewner's conjecture

Naoya ANDO

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Abstract

Let f be a smooth function of two variables x , y and for each positive integer n , let d n f be a symmetric tensor field of type ( 0 , n ) defined by d n f : = i = 0 n n i x n - i y i f d x n - i d y i and 𝒟 ˜ d n f a finitely many-valued one-dimensional distribution obtained from d n f : for example, 𝒟 ˜ d 1 f is the one-dimensional distribution defined by the gradient vector field of f ; 𝒟 ˜ d 2 f consists of two one-dimensional distributions obtained from one-dimensional eigenspaces of Hessian of f . In the present paper, we shall study the behavior of 𝒟 ˜ d n f around its isolated singularity in ways which appear in [1]--[4]. In particular, we shall introduce and study a conjecture which asserts that the index of an isolated singularity with respect to 𝒟 ˜ d n f is not more than one.

Article information

Source
J. Math. Soc. Japan, Volume 57, Number 1 (2005), 1-20.

Dates
First available in Project Euclid: 13 October 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1160745810

Digital Object Identifier
doi:10.2969/jmsj/1160745810

Mathematical Reviews number (MathSciNet)
MR2114717

Zentralblatt MATH identifier
1071.53002

Subjects
Primary: 37E35: Flows on surfaces
Secondary: 53A05: Surfaces in Euclidean space 53B25: Local submanifolds [See also 53C40]

Keywords
Loewner's conjecture the index conjecture Carathéodory's conjecture symmetric tensor field critical direction umbilical point many-valued one-dimensional distribution index

Citation

ANDO, Naoya. A conjecture in relation to Loewner's conjecture. J. Math. Soc. Japan 57 (2005), no. 1, 1--20. doi:10.2969/jmsj/1160745810. https://projecteuclid.org/euclid.jmsj/1160745810


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