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Abstract
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.
Highlights
One area in which linear and nonlinear equations appear to be in very close relationship is the embedding of Riemannian manifolds into manifolds of higher dimension
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
Summary
One area in which linear and nonlinear equations appear to be in very close relationship is the embedding of Riemannian manifolds into manifolds of higher dimension. The embedded manifold is constructed by means of linear differential equations. These equations form an overdetermined set and the integrability conditions they obey in order for a solution to exist are in general nonlinear differential equations. They would be obeyed by the metric or second fundamental form of the embedded manifold, for example
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