## Journal of the Mathematical Society of Japan

### A conjecture in relation to Loewner's conjecture

Naoya ANDO

#### 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 $d^n f:=\sum^n_{i=0}$ $\left(\begin{array}{c}n\\i\end{array}\right)$ $\left( \partial^{n-i}_x \partial^i_y f\right) dx^{n-i} dy^i$ and $\tilde{\mathscr D}_{d^n f}$ a finitely many-valued one-dimensional distribution obtained from $d^n f$: for example, $\tilde{\mathscr D}_{d^1 f}$ is the one-dimensional distribution defined by the gradient vector field of $f$; $\tilde{\mathscr D}_{d^2 f}$ 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 $\tilde{\mathscr D}_{d^n f}$ 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 $\tilde{\mathscr D}_{d^n f}$ 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]

#### 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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