伪黎曼曲面之间的谐波映射

IF 1.6 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics Pub Date : 2024-10-09 DOI:10.1016/j.geomphys.2024.105340

A. Fotiadis, C. Daskaloyannis

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引用次数: 0

摘要

我们研究黎曼曲面或洛伦兹曲面M与黎曼曲面或洛伦兹曲面N之间的局部谐波映射。众所周知，黎曼曲面之间的谐波映射是通过正弦-戈登方程的解的分类来划分的。我们将这一结果推广到黎曼曲面或伪黎曼曲面之间谐波映射的所有四种情况。我们研究了该方程的单孑子解，并以统一的方式找到了相应的调和映射。接下来，我们讨论了调和映射方程的贝克隆变换，它提供了两个正弦或正弦-哥顿型方程的解之间的联系。最后，我们举例说明利用贝克隆变换构建的谐波图。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Harmonic maps between pseudo-Riemannian surfaces

We study locally harmonic maps between a Riemann surface or a Lorentz surface M and a Riemann or Lorentz surface N. All four cases are written using a unified formalism. Therefore properties and solutions to the harmonic map problem can be studied in a unified way.

It is known that harmonic maps between Riemannian surfaces are classified by the classification of the solutions of a sinh-Gordon equation. We extend this result to all the four cases of harmonic maps between Riemannian or pseudo-Riemannian surfaces. The calculation of the corresponding harmonic map can be calculated by the solutions of the corresponding Beltrami equations in all the cases.

We study the one-soliton solutions of this equation and we find the corresponding harmonic maps in a unified way.

Next, we discuss a Bäcklund transformation of the harmonic map equations that provides a connection between the solutions of two sine or sinh-Gordon type equations. Finally, we give an example of a harmonic map that is constructed by the use of a Bäcklund transformation.

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来源期刊

Journal of Geometry and Physics 物理-物理：数学物理

CiteScore

2.90

自引率

6.70%

发文量

205

审稿时长

64 days

期刊介绍： The Journal of Geometry and Physics is an International Journal in Mathematical Physics. The Journal stimulates the interaction between geometry and physics by publishing primary research, feature and review articles which are of common interest to practitioners in both fields. The Journal of Geometry and Physics now also accepts Letters, allowing for rapid dissemination of outstanding results in the field of geometry and physics. Letters should not exceed a maximum of five printed journal pages (or contain a maximum of 5000 words) and should contain novel, cutting edge results that are of broad interest to the mathematical physics community. Only Letters which are expected to make a significant addition to the literature in the field will be considered. The Journal covers the following areas of research: Methods of: • Algebraic and Differential Topology • Algebraic Geometry • Real and Complex Differential Geometry • Riemannian Manifolds • Symplectic Geometry • Global Analysis, Analysis on Manifolds • Geometric Theory of Differential Equations • Geometric Control Theory • Lie Groups and Lie Algebras • Supermanifolds and Supergroups • Discrete Geometry • Spinors and Twistors Applications to: • Strings and Superstrings • Noncommutative Topology and Geometry • Quantum Groups • Geometric Methods in Statistics and Probability • Geometry Approaches to Thermodynamics • Classical and Quantum Dynamical Systems • Classical and Quantum Integrable Systems • Classical and Quantum Mechanics • Classical and Quantum Field Theory • General Relativity • Quantum Information • Quantum Gravity