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Rotational surfaces in a normed 3-space whose principal curvatures satisfy a linear relation

Makoto Sakaki, Kakeru Yanase

Abstract

We classify rotational surfaces in a normed 3-space with rotationally symmetric norm whose principal curvatures satisfy a linear relation.

Rotational surfaces in a normed 3-space whose principal curvatures satisfy a linear relation

Abstract

We classify rotational surfaces in a normed 3-space with rotationally symmetric norm whose principal curvatures satisfy a linear relation.
Paper Structure (10 sections, 6 theorems, 281 equations)

This paper contains 10 sections, 6 theorems, 281 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rotational surfaces
  4. The case where one of principal curvatures is constant
  5. Homogeneous linear Weingarten cases
  6. Inhomogeneous linear Weingarten cases
  7. The case where $\lambda = -1$
  8. Connecting argument for $\lambda = -1$
  9. The case where $\lambda \neq -1$
  10. Connecting argument for $\lambda \neq -1$

Key Result

Theorem 5.1

A rotational surface in $({\mathbb R}^3, \|\cdot\|)$ given by where $\alpha > 0$ and $u' \neq 0$, satisfies $k_1+\lambda k_2 = 0$ for a non-zero constant $\lambda$, if and only if for a positive constant $c_2$.

Theorems & Definitions (6)

  • Theorem 5.1
  • Theorem 5.2
  • Theorem 5.3
  • Theorem 5.4
  • Theorem 6.1
  • Theorem 6.2