A conjecture in relation to Loewner's conjecture

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概要

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• 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:=∑^n_{i=0} n \atopwithdelims() i \big( ∂<SUP>n-i</SUP>_x ∂^i_y f\big) dx<SUP>n-i</SUP> dy^i and ˜{\cal D}_{d^n f} a finitely many-valued one-dimensional distribution obtained from d^n f: for example, ˜{\cal D}_{d^1 f} is the one-dimensional distribution defined by the gradient vector field of f; ˜{\cal 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 ˜{\cal 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 ˜{\cal D}_{d^n f} is not more than one.

• 社団法人 日本数学会の論文
• 2005-01-01

著者

• Ando Naoya

  Faculty Of Science Kumamoto University

関連論文

• A TWO-DIMENSIONAL RIEMANNIAN MANIFOLD WITH TWO ONE-DIMENSIONAL DISTRIBUTIONS
• A conjecture in relation to Loewner's conjecture

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