By Thurston W.P.

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Within the Spring of 1966, I gave a sequence of lectures within the Princeton college division of Physics, aimed toward contemporary mathematical ends up in mechanics, particularly the paintings of Kolmogorov, Arnold, and Moser and its program to Laplace's query of balance of the sunlight process. Mr. Marsden's notes of the lectures, with a few revision and growth by means of either one of us, grew to become this booklet.

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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.