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January, 2005 A conjecture in relation to Loewner's conjecture
Naoya ANDO
J. Math. Soc. Japan 57(1): 1-20 (January, 2005). DOI: 10.2969/jmsj/1160745810


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.


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Naoya ANDO. "A conjecture in relation to Loewner's conjecture." J. Math. Soc. Japan 57 (1) 1 - 20, January, 2005.


Published: January, 2005
First available in Project Euclid: 13 October 2006

zbMATH: 1071.53002
MathSciNet: MR2114717
Digital Object Identifier: 10.2969/jmsj/1160745810

Primary: 37E35
Secondary: 53A05 , 53B25

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

Rights: Copyright © 2005 Mathematical Society of Japan


Vol.57 • No. 1 • January, 2005
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