Abstract
AbstractThis final chapter discusses how the main definitions and results need to be modified in the semi-Riemannian case. In this context, harmonic maps include the strings of mathematical physics. Weakly conformal and horizontally weakly conformal maps are discussed; care is taken with the definitions as the subspaces of the tangent spaces involved may be degenerate. It is shown that with appropriate definitions, the characterization of harmonic morphisms as horizontally weakly conformal harmonic maps carries over to the semi-Riemannian case. Certain harmonic morphisms are simply null solutions of the wave equation. The chapter concludes with an explicit local description of all harmonic morphisms between Lorentzian 2-manifolds. In the ‘Notes and comments’ section, the connection with the shear-free ray congruences of mathematical physics is described.
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