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AIMS Mathematics

2016, Volume 1, Issue 1: 1-8. doi: 10.3934/Math.2016.1.1
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Research article

Harmonic Maps Surfaces and Relativistic Strings

  • Department of Mathematics, University of Texas, Edinbrug, TX, 78540-2999, USA
  • Received: 26 March 2015 Accepted: 06 April 2016 Published: 14 April 2016

  • The harmonic map is introduced and several physical applications are presented. The classical nonlinear σ model can be looked at as the embedding of a two-dimensional surface in a threedimensional sphere, which is itself embedded in a four-dimensional space. A system of nonlinear evolution equations are obtained by working out the zero curvature condition for the Gauss equations relevant to this geometric formulation.

    Citation: Paul Bracken. Harmonic Maps Surfaces and Relativistic Strings[J]. AIMS Mathematics, 2016, 1(1): 1-8. doi: 10.3934/Math.2016.1.1

    Related Papers:

  • The harmonic map is introduced and several physical applications are presented. The classical nonlinear σ model can be looked at as the embedding of a two-dimensional surface in a threedimensional sphere, which is itself embedded in a four-dimensional space. A system of nonlinear evolution equations are obtained by working out the zero curvature condition for the Gauss equations relevant to this geometric formulation.


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    [1] P. Bracken, A. M. Grundland, On Certain Classes of Solutions of the Weierstrass-Enneper System Inducing Constant Mean Curvature Surfaces, J. Nonlin. Math. Phys. 6 (1999), 294-313.
    [2] P.Bracken, A. M. Grundland, Properties and Explicit Solutions of the Generalized Weierstrass System, J. Math. Phys. 42 (2001), 1250-1282.
    [3] S. Helgason, Di erential Geometry and Symmetric Spaces, Academic Press, New York, 1962.
    [4] F. Lund, T. Regge, Unified Approach to Strings and Vortices with Soliton Solutions, Phys. Rev. D, 14 (1976), 1524.
    [5] F. Lund, Note on the Geometry of the Nonlinear σ Model in Two Dimensions, Phys. Rev. D, 15 (1977), 1540-1543.
    [6] C. W. Misner, Harmonic maps as models for physical theories, Phys. Rev. D, 18 (1978), 4510-4524.
    [7] K. Pohlmeyer, Integrable Hamiltonian Systems and Interactions through Quadratic Constraints, Commun. Math. Phys. 46 (1976), 207-221.
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  • © 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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