This 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.

Harmonic morphisms are mappings between Riemannian manifolds which preserve Laplace's equation. They can be characterized as harmonic maps which enjoy an extra property called horizontal weak conformality or semiconformality. We shall give a brief survey of the theory concentrating on (i) twistor methods, (ii) harmonic morphisms with one-dimensional fibres; in particular we shall outline the connections with two equations of Mathematical Physics: the monopole equation and the Beltrami fields equation of hydrodynamics.

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We study 3-dimensional Ricci solitons which project via a semi-conformal mapping to a surface. We reformulate the equations in terms of parameters of the map; this enables us to give an ansatz for constructing solitons in terms of data on the surface. A complete description of the soliton structures on all the 3-dimensional geometries is given, in particular, non-gradient solitons are found on Nil and Sol.