{"title":"伪黎曼曲面之间的谐波映射","authors":"A. Fotiadis, C. Daskaloyannis","doi":"10.1016/j.geomphys.2024.105340","DOIUrl":null,"url":null,"abstract":"<div><div>We study locally harmonic maps between a Riemann surface or a Lorentz surface <em>M</em> and a Riemann or Lorentz surface <em>N</em>. 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.</div><div>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.</div><div>We study the one-soliton solutions of this equation and we find the corresponding harmonic maps in a unified way.</div><div>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.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":null,"pages":null},"PeriodicalIF":1.6000,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Harmonic maps between pseudo-Riemannian surfaces\",\"authors\":\"A. Fotiadis, C. Daskaloyannis\",\"doi\":\"10.1016/j.geomphys.2024.105340\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We study locally harmonic maps between a Riemann surface or a Lorentz surface <em>M</em> and a Riemann or Lorentz surface <em>N</em>. 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.</div><div>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.</div><div>We study the one-soliton solutions of this equation and we find the corresponding harmonic maps in a unified way.</div><div>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.</div></div>\",\"PeriodicalId\":55602,\"journal\":{\"name\":\"Journal of Geometry and Physics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.6000,\"publicationDate\":\"2024-10-09\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Geometry and Physics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0393044024002419\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Geometry and Physics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0393044024002419","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.
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
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