Riemann and the Cutting of Surfaces
Shortly after Cauchy and Puiseux, Riemann came up with a radically different solution to the problem of multivaluedness of functions.
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Riemann and the Cutting of Surfaces
Shortly after Cauchy and Puiseux, Riemann came up with a radically different solution to the problem of multivaluedness of functions. He started with the same thought experiment as Cauchy, that of analytic continuation along a loop. What changed is that he imagined a very thin “sheet” propagating along the loop and such that, when one comes back to the starting point, one arrives on a different sheet whenever the value of the function obtained by analytic continuation is different from its initial value. In this way, by performing the analytic continuation along all the possible loops, one associates to the given algebraic function a many sheeted smooth compact surface which covers the Riemann sphere C D C [ f1g associated with the plane C of one complex variable x. This surface was later called “the Riemann surface associated with the multivalued function y.x/”. Riemann introduced this construction as early as 1851, in [160], but it was in his 1857 paper [161, Preliminaries, I] where he described it in the most visual way1 : In several investigations, for instance in the study of algebraic and abelian functions, it will be useful to represent the way in which a multivalued function ramifies, in the following geometric way: Conceive a surface extended along the .x; y/ plane and coinciding with it (or, if one wants, an infinitely thin body covering the plane), which extends only so far as the function does. When the function extends, this surface will also extend with it. In a region of the plane where two or more extensions of the function occur, the surface will be double or multiple. It will consist of two or several sheets, each one of them corresponding to a branch of the function. Around a ramification point of the function, a sheet of the surface will extend to another sheet, and in such a way that, in the neighborhood of this point, the surface may be imagined as a helicoid whose axis is perpendicular to the .x; y/ plane at that point, and whose thread is infinitely small. But when, after several turns of z [D x C iy] around the ramification value, the function takes back its initial value, one must assume that the superior sheet of the surface connects to the inferior one by traversing the rest of the sheets.
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In the next fragment, x and y denote the real and imaginary parts of the variable z 2 C.
© Springer International Publishing Switzerland 2016 P. Popescu-Pampu, What is the Genus?, Lecture Notes in Mathematics 2162, DOI 10.1007/978-3-319-42312-8_14
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14 Riemann and the Cutting of Surfaces
Fig. 14.1 The self-intersection has to be neglected!
At each point of a surface which represents the way it ramifies, the multivalued function admits a single determined value, and may therefore be looked upon as a perfectly determined function of the place (of a point) on this surface.
This last sentence is essential: therefore, the function which was multivalued in terms of the variable z becomes univalued as a function defined on the associated surface. If one wan
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