A TWO-DIMENSIONAL RIEMANNIAN MANIFOLD WITH TWO ONE-DIMENSIONAL DISTRIBUTIONS

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概要

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• Let $ M $ be a two-dimensional Riemannian manifold with nowhere zero curvature and let $ \mathcal{D_1}, \mathcal{D_2} $ be two smooth one-dimensional distributions on M orthogonal to each other at any point. We present a method of seeing whether M may be locally and isometrically immersed in $ \mathbb{R}^3 $ so that $ \mathcal{D_1} $ and $ \mathcal{D_2} $ give principal distributions, and in the case where $ M $ may be immersed in such a manner, we specify the values that may become principal curvatures at each point of $ M $. In addition, we study relations among the first fundamental form, the principal distributions and the principal curvature functions on each of a parallel curved surface and a surface with constant mean curvature in $ \mathbb{R}^3 $.

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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