By Paul Baird, John C. Wood

This is often the 1st account in e-book kind of the idea of harmonic morphisms among Riemannian manifolds. Harmonic morphisms are maps which look after Laplace's equation. they are often characterised as harmonic maps which fulfill an extra first order situation. Examples contain harmonic services, conformal mappings within the aircraft, and holomorphic services with values in a Riemann floor. There are connections with many conepts in differential geometry, for instance, Killing fields, geodesics, foliations, Clifford structures, twistor areas, Hermitian buildings, iso-parametric mappings, and Einstein metrics and likewise the Brownain pathpreserving maps of chance conception. Giving an entire account of the basic features of the topic, this ebook is self-contained, assuming just a easy wisdom of differential geometry.

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1, this generalizes to any Riemannian manifold as follows. 2) for any C2 real-valued function f defined on an open subset U of M. The equation A f = 0 is called Laplace's equation and solutions are called harmonic functions (on U). In terms of an orthonormal frame lei} on M, Af = E{ei(ei(f)) - (V ei) f}. 3) Riemannian manifolds and conformality 36 where IgI = det(gk(), so that Laplace's equation reads a (119) (0r z = 0; equivalently, xiaxi - ax 0. OX -2 . Because of this simple formula, the Laplacian satisfies many identities familiar in the case of R.

On U2 the argument of JxJ 2 - 1 + 2ix1 varies continuously in the range (-ir, 7r) and so, for each choice of sign, we get a smooth solution p2 : U2 --* S2 which agrees with cp} only in the upper halfspace R = {(x1i x2, x3) : x1 > 0} . Note that, in contrast to cpi , the maps cp2 are surjective. We call cp2 the (outer) disc example. The fibres of cp2 consist of the half lines given by (i) the intersection of the lines f+ with the upper half-space R, (ii) the intersection of the lines e_ with the lower half-space x1 < 0, and (iii) the tangent half-lines starting at a point of the unit circle.

1). Even if M is not oriented, we can regard it locally as a Riemann surface and use complex notation. Thus, suppose that M is a Riemann surface and let z = x + iy be a complex coordinate so that, as before, z = x - iy. 6 for higher dimensions-for a two-dimensional domain, the converse is true locally as follows. 8 Let f : M -* R be a harmonic function on a Riemann surface. Then on any simply connected domain of M, f is the real part of a holomorphac function. 12) and 8x ay ay ax Consider the 1-form 0 = - (Of /ay) dx + (Of /ax) dy.