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On orthogonal curvilinear coordinate systems in constant curvature spaces

Dmitry Berdinsky, Ivan Rybnikov

Abstract

We describe a method for constructing $n$-orthogonal coordinate systems in constant curvature spaces. The construction proposed is a modification of Krichever's method for producing orthogonal curvilinear coordinate systems in the $n$-dimensional Euclidean space. To demonstrate how this method works, we construct examples of orthogonal coordinate systems on the two-dimensional sphere and the hyperbolic plane, in the case when the spectral curve is reducible and all irreducible components are isomorphic to a complex projective line.

On orthogonal curvilinear coordinate systems in constant curvature spaces

Abstract

We describe a method for constructing -orthogonal coordinate systems in constant curvature spaces. The construction proposed is a modification of Krichever's method for producing orthogonal curvilinear coordinate systems in the -dimensional Euclidean space. To demonstrate how this method works, we construct examples of orthogonal coordinate systems on the two-dimensional sphere and the hyperbolic plane, in the case when the spectral curve is reducible and all irreducible components are isomorphic to a complex projective line.

Paper Structure

This paper contains 4 sections, 4 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Multipoint Baker--Akhiezer functions
  3. Orthogonal coordinate systems for $S^n$ and $H^n$
  4. Orthogonal coordinate systems for $S^n$ and $H^n$ corresponding to singular curves. Examples.

Key Result

Theorem 1

Assume that there exists a 1--form $\Omega$ such that We use the following notation: $A_i = \operatorname{Res}_{Q_i} \Omega$, $B = \operatorname{Res}_{r} \Omega$ and $C_i$ is defined from the expansion $( -C_i k_i ^{-1} + \dots ) dk_i ^{-1}$ of the form $\Omega$ in a neighborhood of $P_i$, $i=1, \dots,u_n$. Let $h = \psi (r)$ be the constant of normali

Figures (2)

  • Figure 2: The coordinate lines $u=const$ and $v=const$ on $S^2$.
  • Figure 3: The coordinate lines $u=const$ and $v=const$ on $H^2$.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • Corollary 1
  • Corollary 2